EP3213033B1 - Method of estimating a navigation state constrained in terms of observability - Google Patents

Description

GENERAL AREA

The invention relates to the field of inertial navigation units.

The invention more particularly relates to a method for estimating the navigation state of a mobile carrier using an extended Kalman filter (generally abbreviated as EKF in the literature), as well as an inertial unit adapted to implement such a device. process.

STATE OF THE ART

An inertial navigation unit is an apparatus for providing real-time information on the state of a carrier: position, speed, orientation, etc. This information can be used to drive the wearer.

A known solution for estimating them consists in implementing in an inertial unit an extended Kalman filter which provides an estimated state of the carrier by merging data of inertial sensors (accelerometers, gyrometers), and of non-inertial sensors adapted to the type of carrier. (odometer, baro-altimeter, Doppler radar, GPS receiver, etc.). Inertial data play the role of commands, and non-inertial data that of observations.

The Kalman filter is an algorithm typically comprising a step of prediction and a step of updating repeated in time: the prediction step calculates a current state from a previous state and input commands; the update step refines the current state using observations. Prediction calculations are based on a propagation equation modeling the dynamics of state versus state and control. The update calculations are based on an observation equation modeling the observation as a function of the state.

In the Kalman filter, the propagation and observation equations are linear and take the form of matrices that do not depend on the estimated state.

In the extended Kalman filter, at least one of these functions is nonlinear. The data fusion is obtained by linearizing these functions at the steps spread and update. Calculations are made using matrices that depend on the estimated state.

The extended Kalman filter can then be considered as an algorithm performing on the one hand non-linear calculations and on the other hand linear calculations performed in a Kalman filter inside the extended Kalman filter. The state predicted by the nonlinear propagation equation will in the following be called "global state". The state estimated by the Kalman filter will be named "linearized state". The global state can be corrected by the information carried by the linearized state during resetting steps.

A Kalman filter (extended or not) also calculates a margin of error associated with the estimation of the state taking into account an estimation of the statistical error due to many factors such as initial errors and errors. noise of the inertial sensors used. The statistical error at a given instant can be represented in the form of a cloud of points in the state space, each point corresponding to a possible realization. To estimate this margin of error, the power station has a Gaussian statistical model of the error characterized at each instant by a zero average, and a covariance matrix. The latter defines the envelope of an ellipsoidal volume in the state space, centered on the null error, and containing 99.97% of the estimated errors in each direction of the state space. This envelope is subsequently called "envelope 3 sigma".

The true statistical law of errors is unknown and characterized by another 3 sigma envelope. In general, this envelope is not centered on zero and its shape is non-ellipsoidal.

The figure 1 represents the evolution of an estimated 3 sigma envelope and a true sigma 3 envelope at different iterations of the Kalman filter. The true sigma 3 envelope must be contained in the estimated 3 sigma envelope. This constitutes a coherence criterion that must be respected at all times and for all the realizations of the random variables of the system.

In general, the Kalman filter improves observations over time with its observations, and 3 sigma envelopes tend to flatten the along certain so-called "observable" state space axes, as represented in figure 2 . Axes that are orthogonal to these are said to be "unobservable". The estimated envelope 3 sigma then takes an elongated shape along unobservable axes, which creates correlations between the estimated errors.

Although the imperfection of the model used by the filter can play an important part, it has been found that the nonlinearities and the noise of the system often prevent the criterion of coherence described above from being respected when a Kalman filter extended is used (for example, at time T3 of the figure 2 the true sigma 3 envelope is no longer located within the estimated 3 sigma envelope along an unobservable axis).

In the long term, these inconsistencies can become important and spread to the other axes of the vector space of the state to be estimated.

These inconsistencies generally appear only in particular conditions depending for example on the trajectory of the carrier and the initial conditions. These conditions are specific to observation and can be very difficult to predict. They are all the stronger as the observation is nonlinear.

The cause of the inconsistencies is known: the linearization step, inherent to an extended Kalman filter, is noisy by the estimation errors. Indeed, this linearization is performed at the last estimate produced by the filter, estimated which is by definition an inaccurate data. The second-order changes produced on the linearized functions by this estimation noise then modify the observations and are the cause of the inconsistencies.

In the first case, an observable variable can be considered by the unobservable filter but this case is very unlikely; and in a second case, an unobservable variable can be considered observable by the filter. This reduces the estimated 3 sigma envelope along unobservable axes while it should retain its initial value. However, this does not reduce the true error cloud along these same axes since they are actually unobservable. An increasingly important part of the true error cloud is then likely to be no longer inside the estimated 3 sigma envelope as soon as it begins to reduce oneself. The consequences of this inconsistency can be significant and depend on the initial envelope along unobservable axes in default.

A state-of-the-art solution which in most cases makes it possible to compensate for the reduction in the estimated envelope 3 sigma consists in applying a model noise. Indeed, the EKF algorithm comprises an operation consisting of increasing the covariance in the prediction phase using a model noise matrix in order to take into account the variables and their dynamics that have not been modeled in the filter. In this coherence problem, the noise matrix is diverted from its function because the reduction of the estimated envelope is not due to a modeling problem but to an observability problem. Moreover, for reasons of architecture and numerical precision of computation, the noises are often applied on the diagonal of the matrix of noises of model: Like the matrix of covariance, the diagonal of this matrix corresponds to the components of the linearized state followed by the filter. As a result, noise is not applied only in the direction of unobservable axes. As a result, the problem of coherence may not be completely solved, and moreover it may affect the accuracy of the estimates.

Indeed, the applied noise includes a projection on unobservable axes and a projection on observable axes, these projections being able to evolve over time according to the evolution of the unobservable axes. The larger the projection according to the observable axes, the greater the asymptotic value of the error according to these directions, and therefore the accuracy of the estimates is good.

The value of the noise to be applied as for it is not known. An empirical value is chosen, usually large to account for all known problem cases. This aggravates the problem of precision.

On the other hand, as the pattern noise is not applied in the right direction, the estimated envelope elongates in a direction away from the true unobservable axis. As a result, this ellipsoidal envelope, which certainly becomes larger along its main axes, may no longer cover the cloud with true errors. It is therefore likely to remain an inconsistency.

The problem of observability is thus treated according to the state of the art by a solution which does not completely solve the problem, and which furthermore hinders the accuracy of the estimates.

It is also known from the document WO 2014/130854 a method of estimating a state using an extended Kalman filter EKF. This EKF filter is subject to observability constraints; the transition matrix and the observation matrix used by this EKF filter are adjusted.

PRESENTATION OF THE INVENTION

• acquisition de mesures d'au moins une des variables,
• filtrage de Kalman étendu produisant un état estimé courant et une matrice de covariance délimitant dans l'espace de l'état de navigation une région d'erreurs, à partir d'un état estimé précédent, d'une matrice d'observation, d'une matrice de transition et des mesures acquises,

le procédé étant caractérisé en ce qu'il comprend une étape d'ajustement de la matrice de transition et de la matrice d'observation avant leur utilisation dans le filtrage de Kalman étendu, de sorte que les matrices ajustées vérifient une condition d'observabilité qui dépend d'au moins une des variables de l'état du porteur, la condition d'observabilité étant adaptée pour empêcher le filtre de Kalman de réduire la dimension de la région le long d'au moins un axe non-observable de l'espace d'état,
dans lequel la condition d'observabilité à vérifier par les matrices de transition et d'observation ajustées est la nullité du noyau d'une matrice d'observabilité qui leur est associée, et
dans lequel l'ajustement comprend les étapes de :

• calcul d'au moins une base primaire de vecteurs non-observables à partir de l'état estimé précédent,
• pour chaque matrice à ajuster, calcul d'au moins un écart matriciel associé à la matrice à partir de la base primaire de vecteurs,

décalage de chaque matrice à ajuster selon l'écart matriciel qui lui est associé de sorte à vérifier la condition d'observabilité.The aim of the invention is to estimate the global state of a mobile carrier governed by a non-linear model, while minimizing the appearance of the inconsistencies defined in the introduction.
It is therefore proposed a method for estimating a multi-variable navigation state of a mobile carrier according to the extended Kalman filter method, comprising the steps of:

• acquiring measurements of at least one of the variables,
• extended Kalman filtering producing a current estimated state and a covariance matrix defining in the navigation state space a region of errors, from a previous estimated state, of an observation matrix, a transition matrix and acquired measures,

the method being characterized in that it comprises a step of adjusting the transition matrix and the observation matrix before their use in the extended Kalman filtering, so that the adjusted matrices verify an observability condition which depends on at least one of the variables of the carrier state, the observability condition being adapted to prevent the Kalman filter from reducing the dimension of the region along at least one unobservable axis of the space state,
in which the condition of observability to be verified by the adjusted transition and observation matrices is the nullity of the core of an observability matrix associated with them, and
wherein the adjustment comprises the steps of:

• calculating at least one primary base of unobservable vectors from the previous estimated state,
• for each matrix to be adjusted, calculating at least one matrix difference associated with the matrix from the primary vector base,

offset of each matrix to be adjusted according to the matrix gap associated therewith so as to verify the condition of observability.

The method according to the invention not only makes it possible to improve the consistency of the estimates but also makes it possible to improve the precision of the estimates, because the part of model noises used according to the state of the art to solve the problems of observability can be canceled.

Moreover, this method modifies only very little the software architecture of the EKF filter, and can be implemented in the form of a light software update on equipment already in operation. This update consists of adding a 2-matrix adjustment function, and reducing model noise settings.

The invention may also be supplemented by the following features, taken alone or in any of their technically possible combinations.

• propagation de l'état estimé précédent en un état prédit au moyen de la matrice de transition ajustée,
• linéarisation en l'état prédit d'un modèle non-linéaire pour produire la matrice d'observation avant ajustement,
• ajustement de la matrice d'observation produite par linéarisation.

The extended Kalman filtering step may include the substeps of:

• propagating the previous estimated state to a predicted state using the adjusted transition matrix,
• linearization in the predicted state of a non-linear model to produce the observation matrix before adjustment,
• adjustment of the observation matrix produced by linearization.

Implementing the linearization after the propagation step makes it possible for this linearization to be performed at a point in the vector space that is more likely to exist: the predicted state produced by the propagation step. Therefore, the adjustment of the observation matrix takes this propagation into account, and provides more accurate results than in the case where linearization is carried out before the propagation step, in the state estimated above.

The adjustment may further comprise a step of orthogonalizing the primary base to a secondary base of vectors, the latter being memorized from one cycle to another, and the matrix gap associated with the observation matrix being calculated from the secondary base of vectors.

The matrix difference associated with the observation matrix may be the sum of several independent matrix deviations, each matrix difference being calculated from a vector of the secondary base which is specific to it.

• mémorisation de la base secondaire de vecteurs et de coefficients d'orthogonalisation qui sont également produits par l'étape d'orthogonalisation, et
• transformation de la base primaire de vecteurs en une base tertiaire de vecteurs au moyen de coefficients d'orthogonalisation mémorisés au cours d'un cycle précédent,
• l'écart matriciel associé à la matrice de transition étant mis en oeuvre à partir d'une base secondaire mémorisée au cours du cycle précédent et de la base tertiaire calculée au cours du cycle courant.

The process steps can be repeated in successive cycles. A given cycle, called current cycle, can then include the steps of:

• storing the secondary vector base and orthogonalization coefficients that are also produced by the orthogonalization step, and
• transforming the primary vector base into a tertiary vector base by means of orthogonalization coefficients stored in a previous cycle,
• the matrix difference associated with the transition matrix being implemented from a secondary base stored during the previous cycle and the tertiary base calculated during the current cycle.

The matrix difference associated with the transition matrix may also be the sum of several independent elementary matrix deviations, each matrix difference being calculated from a secondary vector of the secondary base stored during the previous cycle and from a vector Tertiary tertiary base calculated during the current cycle, the secondary and tertiary vectors being specific to the elementary matrix gap.

• une pluralité d'écarts matriciels candidats, chaque écart matriciel candidat étant calculé à partir d'une base primaire de vecteurs non-observables respective, et
• une pluralité de métriques, chaque métrique étant représentative d'une amplitude d'ajustement induite par un écart matriciel candidat respectif, l'étape de décalage de la matrice à ajuster étant mise en oeuvre selon un écart matriciel élu parmi les écarts matriciels candidats en fonction des métriques calculées.

For at least one matrix to be adjusted, it is also possible to calculate:

• a plurality of candidate matrix gaps, each candidate matrix gap being calculated from a respective primary base of respective unobservable vectors, and
• a plurality of metrics, each metric being representative of an adjustment amount induced by a respective candidate matrix deviation, the step of shifting the matrix to be adjusted being implemented according to a matrix difference elected among the candidate matrix differences as a function of the calculated metrics.

The matrix deviation elected may be the candidate matrix deviation associated with the metric representative of the smallest adjustment amplitude.

The offset can be implemented only if the metric of the matrix gap elected is less than a predetermined threshold.

According to another aspect of the invention, there is provided an inertial unit comprising a plurality of sensors and an estimation module configured to estimate a navigation state of the multi-variable inertial unit by implementing the method according to the first aspect. of the invention.

DESCRIPTION OF THE FIGURES

• Les figures 1 et 2, déjà discutées, représentent deux évolutions d'enveloppes 3 sigma au cours du temps, au cours de la mise en oeuvre d'un filtre de Kalman étendu.
• La figure 3 représente schématiquement un porteur embarquant une centrale inertielle selon un mode de réalisation de l'invention.
• La figure 4 est le schéma fonctionnel d'un estimateur selon un mode de réalisation.
• La figure 5 est le schéma fonctionnel d'un bloc de linéarisation et d'ajustement compris dans l'estimateur de la figure 2.
• La figure 6 est le schéma fonctionnel d'un sous-bloc représenté sur la figure 5.
• La figure 7 illustre une justification géométrique de l'ajustement de matrices utilisées par le filtre de Kalman.
• La figure 8 représente un flux des données correspondant à la mise en oeuvre des étapes illustrées sur la figure 3.

Sur l'ensemble des figures, les éléments similaires portent des références identiques.Other features, objects and advantages of the invention will emerge from the description which follows, which is purely illustrative and nonlimiting, and which should be read with reference to the appended drawings in which:

• The Figures 1 and 2 , already discussed, represent two evolutions of 3 sigma envelopes over time, during the implementation of an extended Kalman filter.
• The figure 3 schematically represents a carrier carrying an inertial unit according to one embodiment of the invention.
• The figure 4 is the block diagram of an estimator according to one embodiment.
• The figure 5 is the block diagram of a linearization and adjustment block included in the estimator of the figure 2 .
• The figure 6 is the block diagram of a sub-block shown on the figure 5 .
• The figure 7 illustrates a geometric justification of the fit of matrices used by the Kalman filter.
• The figure 8 represents a flow of data corresponding to the implementation of the steps illustrated on the figure 3 .

In all the figures, similar elements bear identical references.

DETAILED DESCRIPTION OF THE INVENTION

With reference to the figure 3 , an IN inertial unit is embarked on a mobile carrier P, such as a land vehicle, a helicopter, an airplane, etc.

The inertial unit IN comprises several parts: inertial sensors CI, complementary sensors CC, and means E to implement estimation calculations. These parts can be physically separated from each other.

The inertial sensors CI are typically accelerometers and / or gyrometers respectively measuring specific forces and rotational speeds that the wearer undergoes with respect to an inertial reference system. The specific force corresponds to the acceleration of non-gravitational origin. When these sensors are fixed relative to the carrier, the central is called "strap down" type.

The complementary DC sensors are variable depending on the type of carrier, its dynamics, and the intended application. Inertial units typically use a GNSS receiver (eg GPS). For a land vehicle, it is also one or more odometers. For a boat, it can be a "loch", giving the speed of the boat compared to that of the water or the seabed. Cameras are another example of sensors.

The navigation estimating means E will subsequently be referred to as the "estimator".

The output data of the estimator E are a state of navigation of the carrier and possibly internal states of the inertial unit.

The estimator E comprises in particular an extended EKF Kalman filter configured to merge the information given by the complementary sensors and the inertial sensors so as to provide an optimal estimate of the navigation information.

u(t) représente la commande d'entrée constituée de la force spécifique et de la vitesse angulaire. Ces 2 grandeurs sont mesurées par les capteurs CI de façon discrète dans le temps selon une période T ₁.The fusion is done according to a continuous dynamic system serving as a model to predict the state at each moment by means of a nonlinear propagation function f, as well as the way of observing it with the help of an observation function h which can be non-linear as well (see Appendix 5 recalling some principles on dynamic systems), this function depending on the type of DC sensor used: $$\{\begin{array}{c}\dot{X}\left(t\right)=f\left(X\left(t\right),u\left(t\right)\right)\\ z\left(t\right)=h\left(X\left(t\right)\right)\end{array}$$

u (t) represents the input control consisting of the specific force and the angular velocity. These 2 quantities are measured by the IC sensors discretely in time according to a period T ₁ .

The global state X ( t ) comprises inter alia the coordinates of position, velocity, as well as the orientation of the carrier in the form of one or more rotations, each of them represented for example by a matrix or a quaternion of attitude.

The estimator E exploits a discrete version of these equations with continuous time obtained according to the rules of the state of the art: $$\{\begin{array}{c}{X}_k=\mathrm{boy\; Wut}\left(X_{k-1},u_{k-1}\right)\\ z_k=h\left(X_k\right)\end{array}$$

où φ _k-1→k correspond à la matrice de transition dépendant de la fonction de propagation f, et Q la matrice de bruits de modèle.The EKF filter operates iteratively according to a prediction step taking into account the measurements of the IC sensors and an update step taking into account the measurements of the DC sensors:
The prediction step includes predicting an estimated global state, and predicting an associated covariance.
Predicted global state prediction: X _{k | k -1} = g ( X _{k -1 | k -1} , u _{k -1} )
Associated covariance prediction: ${\widehat{P}}_{k|k-1}={\phi }_{k-1\to k}{\widehat{P}}_{k-1|k-1}{\phi }_{k-1\to k}^T+Q_{k-1}$

where φ _{k- 1 → k} corresponds to the transition matrix depending on the propagation function f, and Q the model noise matrix.

où H_k correspond à la matrice d'observation.
Gain de Kalman : $K_k={\widehat{P}}_{k-1|k-1}{H}_k^T{S}_k^{-1}$

The update step uses the following variables:
Innovation: ỹ _k = z _k - h ( X _{k | k -1} )
Covariance innovation: ${\widehat{S}}_k=H_k{\widehat{P}}_{k-1|k-1}{H}_k^T+R_k$

where H _k corresponds to the observation matrix.
Kalman gain: $K_k={\widehat{P}}_{k-1|k-1}{H}_k^T{S}_k^{-1}$

• Mise à jour état global estimé : X̂_k | _k = X̂ _k|k-1 + K_kỹ_k
• Mise à jour covariance associée : P̂_k | _k = (I - K_kH_k )P̂ _k|k-1
  • Matrice de transition théorique : ${\phi }_{k-1\to k}={\displaystyle {\int }_{\left(k-1\right)T}^{kT}exp}\left(F\left(\tau \right)\right)d\tau$
    
  • Avec : $F\left(t\right)={\frac{\partial f}{\partial X}|}_{X\left(t\right),u\left(t\right)}$
    
  • Matrice d'observation : $H_k={\frac{\partial h}{\partial X}|}_{{\widehat{X}}_{k|k-1}}$
    

The update is implemented as follows:

• Estimated global status update: X _k | _k = X _{k | k -1} + K _k ỹ _k
• Associated covariance update: P _k | _k = ( I - K _k H _k ) P _{k | k -1}
  • Theoretical transition matrix: ${\phi }_{k-1\to k}={\displaystyle {\int }_{\left(k-1\right)T}^{kT}exp}\left(F\left(\tau \right)\right)d\tau$
    
  • With: $F\left(t\right)={\frac{\partial f}{\partial X}|}_{X\left(t\right),u\left(t\right)}$
    
  • Observation matrix: $H_k={\frac{\partial h}{\partial X}|}_{{\widehat{X}}_{k|k-1}}$
    

The estimator E, although close to this algorithm, is slightly different. The updating of the overall state may not be totally additive, in particular to preserve the properties of the rotation matrices, and in particular also to take account of the rapid pace of measurements of the IC sensors. The prediction of the estimated overall state can then be made at a faster rate than the other operations. The difference in rhythm may depend on the dynamics of the wearer. Other differences are possible. For example, an approximation can be made on the transition matrix.

Is illustrated on the figure 4 an estimator E according to an embodiment comprising different functional blocks corresponding to respective steps of its implementation.

The estimation is implemented in successive iterations, each iteration being identified by an index k.

The prediction of the global state estimated according to a rate 1 is carried out in step 100.

A 1-cycle memory and a resetting step operate at the same rate. This registration step takes into account data updated at a rate 2, which is slower.

All other processes operate at rate 2. Step 200 converts rate states 1 into rate states 2. At rate 2, X _{k / k -1} , x _{k / k -1} , X _{k / k} , x _{k / k} , this last state coming from the last update of the Kalman filter 400, represent in the order the predicted global state, the linearized state predicted, the updated global state, the state linearized updated. The state x _{k / k -1} , can in some cases always be 0.

The transition and observation matrices are calculated in the linearization step 300. An adjustment sub-step, specific to the invention and detailed below, is added thereto.

The innovation is calculated from the predicted global state and from the observation of the CC sensors represented at the bottom of the figure 4 .

The other steps of the EKF are grouped in step 400 making estimates using matrices, and constituting a Kalman filter, these estimates relating to linearized states and covariances. A particularity stems from the fact that step 400 does not include a linearized state prediction, which may optionally be incorporated in step 100 at a rate of 1.

Step 401 includes predicting the covariance associated with the predicted global state and exploiting the transition matrix. Step 402 includes calculating the covariance of the innovation, Kalman gains, updating the covariance of the global state, and updating the linearized state. This step exploits the observation matrix.

The linearization points in the step 300 can be calculated as a function of X _{k / k -1} , x _{k / k -1} , X _{k / k} , x _{k / k} , the latter state coming from the last update of the Kalman filter 400. If x _{k / k -1} , = x _{k / k} = 0, then X _{φ, k -1} = X _{k -1 / k -1} and X _{H, k} = X _{k / k -1} .

A first linearization point X _{φ , k -1} makes it possible to calculate in step 300 the transition matrix φ _{k -1 → k} ( X _{φ, k -1} ) at the Kalman period corresponding to the rate 2. This point linearization corresponds to the global state estimated after the last resetting.

A second linearization point X _{H, k} makes it possible to calculate at the Kalman cycle k in step 300 the observation matrix H _k ( X _{H, k} ) . This linearization point is obtained from the calculations of step 100. It corresponds to the last predicted state before the next resetting.

In order to simplify the notations, the transition matrix φ _{k -1 → k} ( X _{φ, k -1} ) will now be denoted by φ _{k -1 → k} , and the observation matrix H _k ( X _{H, k} ) noted now. H _k . φ _{k -1 → k} and H _k are each expressed as a function of an overall state corresponding to their respective linearization points X _{φ, k -1} and X _{H, k} .

As said before, the steps are repeated recursively at the next iteration of time k + 1 at rate 2, rate of the Kalman filter.

Adjustment of transition and observation matrices

The linearization step implemented in each Kalman cycle constitutes a weak point of the estimator E.

Also, the estimation method comprises, in step 300, an additional adjustment step as a function of an observability condition, which adjusts, prior to their use by the Kalman filter, the matrices generated by this linearization step. : On the one hand, the transition matrix used to propagate the covariance matrix, and on the other hand the observation matrix.

The observability condition consists of a constraint imposed on the core of the observability matrix (the definition of this matrix is given in Appendix 6) through an adjustment of the transition and observation matrices so as to include in the core a predetermined model of unobservable vector subspace. This model of non-observability consists of an equation in the state space of a vector subspace represented by a base, according to a global state, this vector subspace being updated at each cycle of Kalman. Including this subspace in the core of the observability matrix makes it unobservable and decreases the risk of inconsistency if the model is relevant.

The global state from which the non-observable rendered vector subspace is calculated corresponds to the linearization point of the observation function (see Annex 6). A non-observable vector has indeed a zero image through the observation matrix, and this vector corresponds to a linearized state in the vicinity of a global state corresponding to the linearization point of the observation function.

During the adjustment step, a corresponding matrix difference will be calculated for each of the two transition and observation matrices.

By convention, stellar notations will be used in the following to designate the matrices of transition and observation obtained by the adjustment.

The flow of the data needed for the adjustment calculation is represented schematically on the figure 8 , highlighting the fact that the adjustment of the observation matrix depends on the linearization point of the observation function in the current Kalman cycle, and that the adjustment of the transition matrix depends on the linearization point of the observation function in the current cycle and in the previous cycle.

It is therefore necessary to exploit a base B _k generating the unobservable vector subspace at cycle k, and a base B _{k -1} generating the unobservable vector subspace at cycle k-1.

It is then possible, on a Kalman cycle, to memorize the linearization point and to generate 2 times per cycle the unobservable base, as it is represented in FIG. X. It is possible in an equivalent way on a Kalman cycle. , to memorize the unobservable base so as to generate it only once per cycle. This second option corresponds to the schematic diagram of the figure 6 . This details the adjustment functions described on the figure 5 .

The figure 6 details the calculations leading to obtaining the two matrix differences relating to the transition and observation matrices at the Kalman cycle of instant k.

Une telle base peut être calculée à chaque cycle de Kalman en utilisant une fonction analytique obtenue lors de la conception du filtre d'après la méthode présentée en Annexe 6 à l'aide d'un système dynamique à temps continu, puis appliquée au système dynamique à temps discret utilisé par la centrale de navigation.The transition matrix of the k-cycle k cycle k-1 cycle and the k-cycle observation matrix are adjusted according to a model of unobservability using the global state vectors X _{H, k -1} and X _{H, k} , linearization points of the observation function at cycles k-1 and k. This model makes it possible to estimate, in a step 301, the unobservable space of the linearized states in the form of an unobservable primary base. ${\mathrm{<figure-callout\; id="1"\; label="B\; k"\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">B\; </figure-callout>}}_k^{}.$

Such a basis can be calculated for each Kalman cycle by using an analytical function obtained in the design of the filter according to the method presented in Annex 6 using a dynamic continuous-time system and then applied to the dynamic system. in discrete time used by the central navigation.

Two examples of risk situations that can be treated according to the method of the invention are described in Appendix 1. An unobservable basic model specific to the static alignment is developed in Appendix 2.

Simplifying assumptions are necessarily made to develop this model during the design phase because no equation containing a limited number of variables can represent the true world. It is possible that the true error cloud is slowly decreasing in all directions of the state space but that the EKF, due to nonlinearities and estimation noise, estimates a faster decay according to certain axes, which creates an inconsistency. It is then possible to model the slow decay along a certain axis by a zero decay in time and to consider this unobservable axis in order to be able to impose a constraint. This simplification slightly reduces accuracy but ensures consistency of estimates. But since it is no longer necessary to resort to model noise to prevent the envelope 3 sigma estimated to be reduced along unobservable axes, it ultimately results in a gain in accuracy.

The static alignment described in Annex 1 is an example of a situation in which the model does not take into account small, imperceptible movements that, in the short term, have no effect, but can be observed in the long term, which causes a slow decay of the envelope 3 true sigma on certain axes. It is proposed, for the sake of simplification of the model, not to take it into account and to model an unobservable base considering that these small movements are null, which is an example of simplification. In the In the second example of Appendix 1, the wave motion can be neglected to model the unobservable base.

est orthogonale, il existe une solution décrite dans l'annexe 3 permettant d'ajuster les matrices de façon simple. Cependant, cette condition, dans le cas général, n'est pas satisfaite. La solution proposée consiste à orthogonaliser la base puis à effectuer l'ajustement. Les traitements décrits ci-dessous sont justifiés par le paragraphe en fin de l'annexe 3 traitant de la base non orthogonale.If the base $B_k^{}$${}_1^{}$

is orthogonal, there is a solution described in Appendix 3 to adjust the matrices in a simple way. However, this condition, in the general case, is not satisfied. The proposed solution is to orthogonalize the base and then perform the adjustment. The treatments described below are justified by the paragraph at the end of Annex 3 dealing with the non-orthogonal basis.

est orthogonalisée en une base secondaire $B_k^2$

selon une variante de la méthode de Gramm-Schmitt, la particularité provenant du fait que les vecteurs ne sont pas rendus unitaires car la normalisation est incluse implicitement dans la formule d'ajustement des matrices présentée par la suite. Cette opération génère une matrice $T_k={\left[a_{i,j,k}\right]}_{\begin{array}{c}1\le i\le q\\ 1\le j\le q\end{array}}$

de coefficients ayant servi à l'orthogonalisation.In an orthogonalization step 302, the primary base $B_k^{}$${}_1^{}$

is orthogonalized to a secondary base $B_k^{}$${}_2^{}$

according to a variant of the Gramm-Schmitt method, the peculiarity coming from the fact that the vectors are not made unitary because the normalization is included implicitly in the formula of adjustment of the matrices presented thereafter. This operation generates a matrix $T_k={\left[{\mathrm{at}}_{i,j,\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">k</figure-callout>}}\right]}_{\begin{array}{c}1\le i\le \mathrm{<figure-callout\; id="1"\; label="q"\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">q</figure-callout>}\\ 1\le j\le q\end{array}}$

of coefficients used for orthogonalization.

et $$B_k^2=\left[\begin{array}{llll}{\widehat{v}}_{\mathrm{2,1,}k}& {\widehat{v}}_{\mathrm{2,2,}k}& \dots & {\widehat{v}}_{\mathrm{2,}q,k}\end{array}\right]$$

Let q be the number of vectors of the unobservable primary base. We pose: $${\mathrm{<figure-callout\; id="1"\; label="B\; k"\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">B\; </figure-callout>}}_k^{}=\left[\begin{array}{cccc}{\widehat{v}}_{\mathrm{1.1,}k}& {\widehat{v}}_{\mathrm{1.2,}k}& ...& {\widehat{v}}_{1q,k}\end{array}\right],$$

and $${\mathrm{<figure-callout\; id="2"\; label="B\; k"\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">B\; </figure-callout>}}_k^{}=\left[\begin{array}{llll}{\widehat{v}}_{\mathrm{2,1,}\mathrm{<figure-callout\; id="2"\; label="k\; v\; ^"\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}"><figure-callout\; id="2"\; label="k\; v\; ^"\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">k\; </figure-callout></figure-callout>}}& {\widehat{v}}_{}{\stackrel{}{}}_{}{\stackrel{}{}}_{\mathrm{2,2,}k}& ...& {\widehat{v}}_{2q,k}\end{array}\right]$$

At cycle k, the orthogonalization step can implement the following calculation: $${\widehat{v}}_{\mathrm{2,1,}k}={\widehat{v}}_{\mathrm{1.1,}k},\mathrm{}{T}_k={\left[0\right]}_{\begin{array}{c}1\le i\le \mathrm{<figure-callout\; id="1"\; label="q"\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">q</figure-callout>}\\ 1\le j\le q\end{array}}$$

Then: ∀ i ≥ 2, i ≤ q : ${\widehat{v}}_{2i,k}={\widehat{v}}_{1i,k}-{\displaystyle \underset{j=1}{\overset{i-1}{\Sigma }}{\mathrm{at}}_{i,j,k}}{\widehat{v}}_{2j,k}$

: coefficients rassemblés dans T_k With ${\mathrm{at}}_{i,j,k}=\{\begin{array}{ll}\underset{\_}{\left({\widehat{v}}_{2j,k}^T{\widehat{v}}_{1i,k}\right)}& \mathrm{if}2\le i\le q\\ \left({\widehat{v}}_{2j,k}^T{\widehat{v}}_{2i,k}\right)& \mathrm{and}1\le j<i\\ 0& \mathrm{if\; not}\end{array}$

: coefficients collected in T _k

ainsi que les coefficients d'orthogonalisation T_k sont mémorisés à l'étape 303 pendant un cycle de Kalman.This base $B_k^{}$${}_2^{}$

and the orthogonalization coefficients T _k are stored in step 303 during a Kalman cycle.

permet de calculer dans une étape 308 un écart matriciel δH_k relatif à la matrice d'observation H_k, et ce sans inversion d'une quelconque matrice, ce qui peut permettre de réduire la charge de calcul. Cette étape peut être omise si la matrice d'observation est indépendante de l'état estimé. Dans ce cas, l'écart matriciel est nul.The base $B_k^{}$${}_2^{}$

allows to calculate in a step 308 a matrix deviation δH _k relative to the observation matrix H _k , without inverting any matrix, which can reduce the calculation load. This step can be omitted if the observation matrix is independent of the estimated state. In this case, the matrix difference is zero.

It is possible to decompose the matrix difference δH _k relative to the observation matrix H _k , into q elementary matrix deviations: $$\mathrm{\delta }{H}_k={\displaystyle \underset{i=1}{\overset{q}{\Sigma }}\mathrm{\delta }{H}_{i,k}}$$

This decomposition thus makes it possible to parallelize the calculation of each elementary matrix difference and thus to shorten the calculation time of step 308.

Each basic matrix deviation can, according to Annex 3, be calculated as follows: For i between 1 and q, $$\delta H_{i,k}=-\frac{1}{{\widehat{v}}_{2i,k}^T{\widehat{v}}_{2i,k}}\left(H_k{\widehat{v}}_{2i,k}\right){\widehat{v}}_{2i,k}^T$$

et les coefficients T _k-1 (mémorisés au cycle de Kalman précédent k-1) permettent de reconstruire une base non observable $B_{k/k-1}^3$

dans une étape de reconstruction référencée 304. Cette base reconstruite à l'instant k dépend de la base primaire à l'instant k, mais aussi des coefficients d'orthogonalisation à l'instant k-1. Son indice est donc noté k/k-1 correspondant à l'instant k sachant les estimations à l'instant k-1. Les bases $B_{k-1}^2$

et $B_{k/k-1}^3$

permettent de calculer l'écart matriciel δφ _k-1→k sans inversion de matrice.Moreover, at cycle k, the base $B_k^{}$${}_1^{}$

and the coefficients T _{k -1} (stored in the previous Kalman cycle k-1) make it possible to reconstruct an unobservable base $B_{k/k-1}^3$

in a reconstruction step referenced 304. This base reconstructed at time k depends on the primary base at time k, but also orthogonalization coefficients at time k-1. Its index is therefore denoted k / k-1 corresponding to the moment k knowing the estimates at time k-1. The basics $B_{k-1}^2$

and $B_{k/k-1}^3$

calculate the matrix difference δφ _{k -1 → k} without matrix inversion.

We pose: $$B_{k/k-1}^3=\left[\begin{array}{llll}{\widehat{v}}_{\mathrm{3,1,}k/k-1}& {\widehat{\mathrm{<figure-callout\; id="3"\; label="v\; ^"\; filenames="imgf0001.png"\; state="\{\{state\}\}"><figure-callout\; id="2"\; label="v\; ^"\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">v\; </figure-callout></figure-callout>}}}_{\mathrm{3,2,}k/k-1}& \cdots & {\widehat{v}}_{3q,k/k-1}\end{array}\right]$$

At cycle k, the reconstruction step 304 may implement the following calculation:

$${\widehat{v}}_{\mathrm{3,1,}k/k-1}={\widehat{v}}_{\mathrm{1,1,}k}$$

Coefficients α _{j, k -1} extracted from $T_{k-1}={\left[{\mathrm{at}}_{i,j,k-1}\right]}_{\begin{array}{c}1\le i\le \mathrm{<figure-callout\; id="1"\; label="q"\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">q</figure-callout>}\\ 1\le j\le q\end{array}}$

$${\widehat{v}}_{\mathrm{3,1,}k/k-1}={\widehat{v}}_{\mathrm{1.1,}k}$$

Then: ∀ i ≥ 2, i ≤ q : ${\widehat{v}}_{3i,k/k-1}={\widehat{v}}_{1i,k}-{\displaystyle \underset{j=1}{\overset{i-1}{\Sigma }}{\mathrm{at}}_{i,j,k-1}}{\widehat{v}}_{3j,k/k-1}$

In a step referenced 306, a matrix difference relating to the transition matrix is calculated.

It is possible to decompose the matrix difference δφ _{i, k -1 → k} relative to the transition matrix φ _{i, k -1 → k} , into q elementary matrix deviations: $$\delta {\phi }_{k-1\to k}={\displaystyle \underset{i=1}{\overset{q}{\Sigma }}\delta {\phi }_{i,k-1\to k}}$$

Each basic matrix deviation may, according to Annex 3, be calculated as follows:
For i between 1 and q, $$\delta {\phi }_{i,k-1\to k}=\frac{1}{{\widehat{v}}_{2i,k-1}^T{\widehat{v}}_{2i,k-1}}\left({\widehat{v}}_{3i,k/k-1}-{\phi }_{k-1\to k}{\widehat{v}}_{2i,k-1}\right){\widehat{v}}_{2i,k-1}^T$$

This decomposition thus makes it possible to parallelize the calculation of each elementary matrix difference and thus can make it possible to shorten the calculation time of step 306.

To reduce memory occupancy and calculations, we can consider only one sub-matrix of the transition matrix: that in correspondence with non-zero values of unobservable vectors.

The two matrix differences obtained make it possible to verify an important condition of observability, a condition which is detailed in Appendix 7.

In addition, Annex 3 explains the origin of the calculation method, and Annex 8 gives a mathematical proof.

The calculation step 306 also produces a metric representative of an adjustment amplitude induced by the matrix difference relative to the transition matrix.

The calculation step 308 also produces a metric representative of an adjustment amplitude induced by the matrix difference relative to the observation matrix.

The same type of metric calculation can be implemented for the transition matrix and the observation matrix. For example, for the transition matrix, one calculates separately the square of the norm of the matrix difference, and that of the transition matrix. The metric then corresponds to the ratio of these two quantities, or to the square root of the ratio of these two quantities.

The standard optionally uses weighting coefficients to normalize the quantities involved in the state vector. For example, a position error may be of the order of a few meters, while an attitude error may be of the order of a few milli-radians. The matrix norm can correspond to the norm of the vector that one would form by putting all the terms of the matrix in the form of a column vector (Froebenius norm).

Appendix 4 explains how to use weights in the calculation of metrics.

The calculation of metrics may use weighting functions g _φ () and g _H (). If the observation matrix H _k is independent of the estimated state, the matrix deviation δH _k is zero, and its metric is zero. It is useless to calculate it. Identically, if the transition matrix is independent of the estimated state, the calculation of the associated metric may be omitted.

Back to the figure 5 In a shifting step 330, the transition matrix is summed to the associated matrix difference, so as to obtain an adjusted transition matrix; and the observation matrix is summed to the associated matrix gap, so as to obtain an adjusted observation matrix.

The sequence 310 illustrated in figure 6 producing a pair of matrix deviations and a pair of metrics can be implemented n times in parallel.

It is assumed that, for n predetermined situations in the continuous domain, we know an unobservable basic model. This model comes from the resolution of a system of equations representative of a respective type of observation and possibly of a respective trajectory of the state vector, a method of which is given in Appendix 6.

In a decision step 320, one of the n matrix difference pairs is elected (for example the pair obtained by the adjustment corresponding to the situation i), and the shifting step is performed only with this pair of dies to produce the adjusted matrices.

The decision step 320 performs a likelihood test on each pair of matrix deviations, by inspecting the value of the corresponding metrics. This test makes it possible to make the decision to recognize this or that situation.

The matrix gap elected is the candidate matrix gap associated with the metric representative of the smallest adjustment amplitude. Thus, the adjustment is made with the most "likely" matrix deviations. The likelihood criterion may relate to the adjustment of the transition matrix and / or the observation matrix.

Furthermore, the shift is implemented only if the metric of the matrix gap elected is less than a predetermined threshold: it may indeed be preferable not to shift the transition and observation matrices, if this offset is not considered sufficiently reliable and introduces additional noise into the system rather than correcting it.

The above estimation method can be implemented by a computer program executed by the calculating means of the inertial unit IN. Furthermore, the identification card 1 comprises a computer program product comprising program code instructions for executing the steps of the method described, when this program is executed by the identification card 1.

ANNEX 1: Examples of risk situations of an inertial navigation unit operating an EKF filter

A first example relates to an inertial navigation unit in the static alignment phase on the ground. In this phase, the local vertical is estimated by measuring the gravity vector using the accelerometers, and the orientation of the plant is estimated by measuring the rotational angular velocity vector of the earth using the gyrometers. The observation used in this phase is typically ZUPT (Zero Velocity Update) which is an observation of a virtual sensor indicating that the speed of the carrier is zero by relation to the Earth. The measurement error is modeled by a white noise. However, inertial sensors sometimes record small non-random movements, for example due to wind or heat expansion effects. This corresponds to a deterministic measurement error not taken into account in the non-inertial sensor model producing the ZUPT observation. There is then an inconsistency between the true dynamic system and the dynamically modeled system. As long as the duration of the static alignment is short, the inconsistency is negligible. However, there are many applications requiring an alignment of several hours, or several days, weeks or months. In this case, it is possible that the inconsistency becomes preponderant.

A second example relates to an inertial navigation unit mounted on a surface vessel using a speed observation projection in the boat mark. This non-inertial measurement is usually done using a "Loch" sensor and is more of a relative measure with respect to water. The fact that it is made in projection in the boat mark creates a dependence of the measure vis-à-vis the attitude of the boat. It is possible then, assuming that only this observation is used, that an inconsistency appears between the estimates of the EKF filter and the true world, and that this inconsistency produces significant errors after several hours if the boat is sailing at constant heading. .

APPENDIX 2: Model of unobservable axes in static alignment

This annex presents the modeling of the unobservable basis for resolving the inconsistency of the first example in Appendix 1 using the proposed method.

Assumptions: The Kalman filter exploits the error model in PHI and the mechanization works in free azimuth.

The following notations are used: [T] Terrestrial landmark. x = axis of rotation [M] Measurement mark [V] Platform marker, in free azimuth T _vt Matrix of passage from [t] to [v] T _vm Matrix of passage from [m] to [v] α Mechanization angle Ω _t Ground rotation speed boy Wut Module of gravity (gravity = gravity ≠ gravitation) g _e Modulus of gravity at the equator a ₀ , K ₁ , K ₂ Constants linked to geodesic constants WGS84 R _x Radius of curvature of the reference ellipsoid in the x direction of [v] R _y Radius of curvature of the reference ellipsoid in the y direction of [v] R _N Radius of curvature of the reference ellipsoid in the North direction R _E Radius of curvature of the reference ellipsoid in the east direction τ Geodetic twist on the reference ellipsoid The Latitude z Altitude Δ x Position error along the x axis of [v] Δ y Position error along the y axis of [v] Δ z Position error along the z axis of [v] Δ V _x Speed error along the x-axis of [v] Δ V _y Speed error along y-axis of [v] Δ V _z Speed error along the z axis of [v] φ _x X-axis rotation error of [v] φ _y Rotation error along y-axis of [v] φ _z Rotation error along the z axis of [v] Δ b _xm Accelerated bias error along the x-axis of [m] Δ b _ym Accelerated bias error along the y axis of [m] Δ b _zm Accelerated bias error along z axis of [m] Δ K _xm Accelerated scale factor error along the x-axis of [m] Δ K _ym Accelerated scale factor error along y-axis of [m] Δ K _zm Accelerated scale factor error along the z axis of [m] Δ d _xm Gyro drift error along the x-axis of [m] Δ d _ym Gyro drift error along the y axis of [m] Δ d _zm Gyro drift error along z axis of [m]

Erreurs de position adns le repère plateforme [v] $$\left[\begin{array}{c}{\phi }_x\\ {\phi }_y\\ {\phi }_z\end{array}\right]$$

Erreurs d'attitude dans le repère plateforme [v] $$\left[\begin{array}{c}\mathrm{\Delta }{b}_{xm}\\ \mathrm{\Delta }{b}_{ym}\\ \mathrm{\Delta }{b}_{zm}\end{array}\right]$$

Erreurs de biais accéléro dans le repère de mesure [m] $$N=\left[\begin{array}{c}\left[\begin{array}{c}\mathrm{\Delta }x\\ \mathrm{\Delta }y\\ \mathrm{\Delta }z\end{array}\right]\\ \left[\begin{array}{c}{\phi }_x\\ {\phi }_y\\ {\phi }_z\end{array}\right]\\ \left[\begin{array}{c}\mathrm{\Delta }{b}_{xm}\\ \mathrm{\Delta }{b}_{ym}\\ \mathrm{\Delta }{b}_{zm}\end{array}\right]\\ \left[\begin{array}{c}\mathrm{\Delta }{K}_{xm}\\ \mathrm{\Delta }{K}_{ym}\\ \mathrm{\Delta }{K}_{zm}\end{array}\right]\\ \begin{array}{l}\left[\begin{array}{c}\mathrm{\Delta }{d}_{xm}\\ \mathrm{\Delta }{d}_{ym}\\ \mathrm{\Delta }{d}_{zm}\end{array}\right]\\ \mathrm{\Delta }\alpha \end{array}\end{array}\right]$$

$$\left[\begin{array}{c}\mathrm{\Delta }{K}_{xm}\\ \mathrm{\Delta }{K}_{ym}\\ \mathrm{\Delta }{K}_{zm}\end{array}\right]$$

Erreurs de facteur d'échelle accéléro dans le repère de mesure [m] $$\left[\begin{array}{c}\mathrm{\Delta }{d}_{xm}\\ \mathrm{\Delta }{d}_{ym}\\ \mathrm{\Delta }{d}_{zm}\end{array}\right]$$

Erreurs de dérive gyro dans le repère de mesure [m] Δα Erreur d'angle de mécanisation en azimuth libre Autres composantes : nulles The state components involved in the equations are presented in an arbitrary order which is as follows: $$\left[\begin{array}{c}\mathrm{\Delta }x\\ \mathrm{\Delta }\mathrm{there}\\ \mathrm{\Delta }z\end{array}\right]$$

Position errors in the platform reference [v] $$\left[\begin{array}{c}{\phi }_x\\ {\phi }_{\mathrm{there}}\\ {\phi }_z\end{array}\right]$$

Attitude errors in the platform reference [v] $$\left[\begin{array}{c}\mathrm{\Delta }{b}_{xm}\\ \mathrm{\Delta }{b}_{\mathrm{there}m}\\ \mathrm{\Delta }{b}_{zm}\end{array}\right]$$

Accelerated bias errors in the measurement mark [m] $$\mathrm{NOT}=\left[\begin{array}{c}\left[\begin{array}{c}\mathrm{\Delta }x\\ \mathrm{\Delta }\mathrm{there}\\ \mathrm{\Delta }z\end{array}\right]\\ \left[\begin{array}{c}{\phi }_x\\ {\phi }_{\mathrm{there}}\\ {\phi }_z\end{array}\right]\\ \left[\begin{array}{c}\mathrm{\Delta }{b}_{xm}\\ \mathrm{\Delta }{b}_{\mathrm{there}m}\\ \mathrm{\Delta }{b}_{zm}\end{array}\right]\\ \left[\begin{array}{c}\mathrm{\Delta }{K}_{xm}\\ \mathrm{\Delta }{K}_{\mathrm{there}m}\\ \mathrm{\Delta }{K}_{zm}\end{array}\right]\\ \begin{array}{l}\left[\begin{array}{c}\mathrm{\Delta }{d}_{xm}\\ \mathrm{\Delta }{d}_{\mathrm{there}m}\\ \mathrm{\Delta }{d}_{zm}\end{array}\right]\\ \mathrm{\Delta }\alpha \end{array}\end{array}\right]$$

$$\left[\begin{array}{c}\mathrm{\Delta }{K}_{xm}\\ \mathrm{\Delta }{K}_{\mathrm{there}m}\\ \mathrm{\Delta }{K}_{zm}\end{array}\right]$$

Accelerometer scale factor errors in the measurement frame [m] $$\left[\begin{array}{c}\mathrm{\Delta }{d}_{xm}\\ \mathrm{\Delta }{d}_{\mathrm{there}m}\\ \mathrm{\Delta }{d}_{zm}\end{array}\right]$$

Gyro drift errors in the measurement mark [m] Δ α Mechanization angle error in free azimuth Other components: null

$$\frac{d}{dt}\left[\begin{array}{c}\mathrm{\Delta }{K}_{xm}\\ \mathrm{\Delta }{K}_{ym}\\ \mathrm{\Delta }{K}_{zm}\end{array}\right]=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]$$

$$\frac{d}{dt}\left[\begin{array}{c}\mathrm{\Delta }{d}_{xm}\\ \mathrm{\Delta }{d}_{ym}\\ \mathrm{\Delta }{d}_{zm}\end{array}\right]=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]$$

Measurements acquired are speeds. The carrier is supposed to be perfectly motionless in translation and rotation. Sensor faults are assumed to be constant over time. accelerometers gyros $$\frac{d}{dt}\left[\begin{array}{c}\mathrm{\Delta }{b}_{xm}\\ \mathrm{\Delta }{b}_{\mathrm{there}m}\\ \mathrm{\Delta }{b}_{zm}\end{array}\right]=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]$$

$$\frac{d}{dt}\left[\begin{array}{c}\mathrm{\Delta }{K}_{xm}\\ \mathrm{\Delta }{K}_{\mathrm{there}m}\\ \mathrm{\Delta }{K}_{zm}\end{array}\right]=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]$$

$$\frac{d}{dt}\left[\begin{array}{c}\mathrm{\Delta }{d}_{xm}\\ \mathrm{\Delta }{d}_{\mathrm{there}m}\\ \mathrm{\Delta }{d}_{zm}\end{array}\right]=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]$$

$$\boxed{\begin{array}{l}\frac{d}{dt}\left[\begin{array}{c}\mathrm{\Delta }{V}_x\\ \mathrm{\Delta }{V}_y\\ \mathrm{\Delta }{V}_z\end{array}\right]=T_{vm}\left[\begin{array}{c}-\mathrm{\Delta }{b}_{xm}\\ -\mathrm{\Delta }{b}_{ym}\\ -\mathrm{\Delta }{b}_{zm}\end{array}\right]+T_{vm}\left[\begin{array}{ccc}-\mathrm{\Delta }{K}_{xm}& 0& 0\\ 0& -\mathrm{\Delta }{K}_{ym}& 0\\ 0& 0& -\mathrm{\Delta }{K}_{zm}\end{array}\right]T_{mv}\left[\begin{array}{c}0\\ 0\\ g\end{array}\right]-\left[\begin{array}{ccc}0& -g& 0\\ g& 0& 0\\ 0& 0& 0\end{array}\right]\left[\begin{array}{c}{\phi }_x\\ {\phi }_y\\ {\phi }_z\end{array}\right]\\ +\left[\begin{array}{ccc}0& -m_{21}& -m_{31}\\ m_{21}& 0& -m_{32}\\ m_{31}& m_{32}& 0\end{array}\right]\left[\begin{array}{c}\mathrm{\Delta }{V}_x\\ \mathrm{\Delta }{V}_y\\ \mathrm{\Delta }{V}_z\end{array}\right]+\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ p_{31}& p_{32}& p_{33}\end{array}\right]\left[\begin{array}{c}\mathrm{\Delta }{V}_x\\ \mathrm{\Delta }{V}_y\\ \mathrm{\Delta }{V}_z\end{array}\right]+\left[\begin{array}{c}{q}_1\\ q_2\\ q_3\end{array}\right]\mathrm{\Delta }\alpha \end{array}}$$

$$\boxed{\frac{d}{dt}\left[\begin{array}{c}{\phi }_x\\ {\phi }_y\\ {\phi }_z\end{array}\right]=T_{vm}\left[\begin{array}{c}\mathrm{\Delta }{d}_{xm}\\ \mathrm{\Delta }{d}_{ym}\\ \mathrm{\Delta }{d}_{zm}\end{array}\right]+A\left(\left[\begin{array}{c}{c}_1\\ c_2\\ c_3\end{array}\right]\right)\left[\begin{array}{c}{\phi }_x\\ {\phi }_y\\ {\phi }_z\end{array}\right]+\left[\begin{array}{c}\frac{1}{\tau }\\ -\frac{1}{R_x}\\ 0\end{array}\right]\mathrm{\Delta }{V}_x\left[\begin{array}{c}\frac{1}{R_y}\\ -\frac{1}{\tau }\\ 0\end{array}\right]\mathrm{\Delta }{V}_y+{\mathrm{\Omega }}_t\left[\begin{array}{c}{a}_1\\ a_2\\ a_3\end{array}\right]\mathrm{\Delta }x+{\mathrm{\Omega }}_t\left[\begin{array}{c}{b}_1\\ b_2\\ b_3\end{array}\right]\mathrm{\Delta }y+{\mathrm{\Omega }}_t\left[\begin{array}{c}-T_{vt}^{21}\\ T_{vt}^{11}\\ 0\end{array}\right]\mathrm{\Delta }\alpha }$$

These equations form a stationary system of the form: $$\boxed{\frac{d}{dt}\left[\begin{array}{c}\mathrm{\Delta }x\\ \mathrm{\Delta }\mathrm{there}\\ \mathrm{\Delta }z\end{array}\right]=\left[\begin{array}{c}\mathrm{\Delta }{V}_x\\ \mathrm{\Delta }{V}_{\mathrm{there}}\\ \mathrm{\Delta }{V}_z\end{array}\right]}$$

$$\boxed{\begin{array}{l}\frac{d}{dt}\left[\begin{array}{c}\mathrm{\Delta }{V}_x\\ \mathrm{\Delta }{V}_{\mathrm{there}}\\ \mathrm{\Delta }{V}_z\end{array}\right]=T_{vm}\left[\begin{array}{c}-\mathrm{\Delta }{b}_{xm}\\ -\mathrm{\Delta }{b}_{\mathrm{there}m}\\ -\mathrm{\Delta }{b}_{zm}\end{array}\right]+T_{vm}\left[\begin{array}{ccc}-\mathrm{\Delta }{K}_{xm}& 0& 0\\ 0& -\mathrm{\Delta }{K}_{\mathrm{there}m}& 0\\ 0& 0& -\mathrm{\Delta }{K}_{zm}\end{array}\right]T_{mv}\left[\begin{array}{c}0\\ 0\\ \mathrm{boy\; Wut}\end{array}\right]-\left[\begin{array}{ccc}0& -\mathrm{boy\; Wut}& 0\\ \mathrm{boy\; Wut}& 0& 0\\ 0& 0& 0\end{array}\right]\left[\begin{array}{c}{\phi }_x\\ {\phi }_{\mathrm{there}}\\ {\phi }_z\end{array}\right]\\ +\left[\begin{array}{ccc}0& -m_{21}& -m_{31}\\ m_{21}& 0& -m_{32}\\ m_{31}& m_{32}& 0\end{array}\right]\left[\begin{array}{c}\mathrm{\Delta }{V}_x\\ \mathrm{\Delta }{V}_{\mathrm{there}}\\ \mathrm{\Delta }{V}_z\end{array}\right]+\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ p_{31}& p_{32}& p_{33}\end{array}\right]\left[\begin{array}{c}\mathrm{\Delta }{V}_x\\ \mathrm{\Delta }{V}_{\mathrm{there}}\\ \mathrm{\Delta }{V}_z\end{array}\right]+\left[\begin{array}{c}{q}_1\\ q_2\\ q_3\end{array}\right]\mathrm{\Delta }\alpha \end{array}}$$

$$\boxed{\frac{d}{dt}\left[\begin{array}{c}{\phi }_x\\ {\phi }_{\mathrm{there}}\\ {\phi }_z\end{array}\right]=T_{vm}\left[\begin{array}{c}\mathrm{\Delta }{d}_{xm}\\ \mathrm{\Delta }{d}_{\mathrm{there}m}\\ \mathrm{\Delta }{d}_{zm}\end{array}\right]+\mathrm{AT}\left(\left[\begin{array}{c}{\mathrm{vs}}_1\\ {\mathrm{vs}}_2\\ {\mathrm{vs}}_3\end{array}\right]\right)\left[\begin{array}{c}{\phi }_x\\ {\phi }_{\mathrm{there}}\\ {\phi }_z\end{array}\right]+\left[\begin{array}{c}\frac{1}{\tau }\\ -\frac{1}{R_x}\\ 0\end{array}\right]\mathrm{\Delta }{V}_x\left[\begin{array}{c}\frac{1}{R_{\mathrm{there}}}\\ -\frac{1}{\tau }\\ 0\end{array}\right]\mathrm{\Delta }{V}_{\mathrm{there}}+{\mathrm{\Omega }}_t\left[\begin{array}{c}{\mathrm{at}}_1\\ {\mathrm{at}}_2\\ {\mathrm{at}}_3\end{array}\right]\mathrm{\Delta }x+{\mathrm{\Omega }}_t\left[\begin{array}{c}{b}_1\\ b_2\\ b_3\end{array}\right]\mathrm{\Delta }\mathrm{there}+{\mathrm{\Omega }}_t\left[\begin{array}{c}-T_{vt}^{21}\\ T_{vt}^{11}\\ 0\end{array}\right]\mathrm{\Delta }\alpha }$$

$\left[\begin{array}{c}\mathrm{\Delta }{b}_{xm}\\ \mathrm{\Delta }{b}_{ym}\\ \mathrm{\Delta }{b}_{zm}\end{array}\right],$

$\left[\begin{array}{c}\mathrm{\Delta }{K}_{xm}\\ \mathrm{\Delta }{K}_{ym}\\ \mathrm{\Delta }{K}_{zm}\end{array}\right],$

$\left[\begin{array}{c}\mathrm{\Delta }{d}_{xm}\\ \mathrm{\Delta }{d}_{ym}\\ \mathrm{\Delta }{d}_{zm}\end{array}\right]$

tels que : $$\forall n\ge \mathrm{0,}\mathrm{}\frac{d^n}{d{t}^n}\left[\begin{array}{c}\mathrm{\Delta }{V}_x\\ \mathrm{\Delta }{V}_y\\ \mathrm{\Delta }{V}_z\end{array}\right]=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]$$

The method of searching for unobservable vectors:
We are looking for $\left[\begin{array}{c}\mathrm{\Delta }{V}_x\\ \mathrm{\Delta }{V}_{\mathrm{there}}\\ \mathrm{\Delta }{V}_z\end{array}\right],$

$\left[\begin{array}{c}\mathrm{\Delta }{b}_{xm}\\ \mathrm{\Delta }{b}_{\mathrm{there}m}\\ \mathrm{\Delta }{b}_{zm}\end{array}\right],$

$\left[\begin{array}{c}\mathrm{\Delta }{K}_{xm}\\ \mathrm{\Delta }{K}_{\mathrm{there}m}\\ \mathrm{\Delta }{K}_{zm}\end{array}\right],$

$\left[\begin{array}{c}\mathrm{\Delta }{d}_{xm}\\ \mathrm{\Delta }{d}_{\mathrm{there}m}\\ \mathrm{\Delta }{d}_{zm}\end{array}\right]$

such as : $$\forall \mathrm{not}\ge 0\mathrm{}\frac{d^{\mathrm{not}}}{d{t}^{\mathrm{not}}}\left[\begin{array}{c}\mathrm{\Delta }{V}_x\\ \mathrm{\Delta }{V}_{\mathrm{there}}\\ \mathrm{\Delta }{V}_z\end{array}\right]=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]$$

$$\left[\begin{array}{c}\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\\ \left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\\ \left[\begin{array}{c}1\\ 0\\ 0\end{array}\right]\\ T_{mv}\left[\begin{array}{c}0\\ -g\\ 0\end{array}\right]\\ \left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\\ T_{mv}\left[\begin{array}{c}0\\ -c_3\\ c_2\end{array}\right]\\ 0\end{array}\right]$$

$$\left[\begin{array}{c}\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\\ \left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\\ \left[\begin{array}{c}0\\ 1\\ 0\end{array}\right]\\ T_{mv}\left[\begin{array}{c}g\\ 0\\ 0\end{array}\right]\\ \left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\\ T_{mv}\left[\begin{array}{c}{c}_3\\ 0\\ -c_1\end{array}\right]\\ 0\end{array}\right]$$

$$\left[\begin{array}{c}\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\\ \left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\\ \left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]\\ \left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\\ \left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\\ T_{mv}\left[\begin{array}{c}-c_2\\ c_1\\ 0\end{array}\right]\\ 0\end{array}\right]$$

$$\left[\begin{array}{c}\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\\ \left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\\ \left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\\ \left[\begin{array}{c}0\\ 0\\ -g{T}_{mv}^{33}\end{array}\right]\\ \left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]\\ \left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\\ 0\end{array}\right]$$

Axe 5 : v ₅ Axe 6 : v ₆ Axe 7 : v ₇ $$\left[\begin{array}{c}\left[\begin{array}{c}\mathrm{\Delta }x\\ \mathrm{\Delta }y\\ \mathrm{\Delta }z\end{array}\right]\\ \left[\begin{array}{c}\mathrm{\Delta }{V}_x\\ \mathrm{\Delta }{V}_y\\ \mathrm{\Delta }{V}_z\end{array}\right]\\ \left[\begin{array}{c}{\mathrm{\phi }}_x\\ {\mathrm{\phi }}_y\\ {\mathrm{\phi }}_z\end{array}\right]\\ \left[\begin{array}{c}\mathrm{\Delta }{b}_{xm}\\ \mathrm{\Delta }{b}_{ym}\\ \mathrm{\Delta }{b}_{zm}\end{array}\right]\\ \left[\begin{array}{c}\mathrm{\Delta }{K}_{xm}\\ \mathrm{\Delta }{K}_{ym}\\ \mathrm{\Delta }{K}_{zm}\end{array}\right]\\ \left[\begin{array}{c}\mathrm{\Delta }{d}_{xm}\\ \mathrm{\Delta }{d}_{ym}\\ \mathrm{\Delta }{d}_{zm}\end{array}\right]\\ \mathrm{\Delta \alpha }\end{array}\right]$$

$$\left[\begin{array}{c}\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right]\\ \left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\\ \left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\\ T_{mv}\left[\begin{array}{c}0\\ 0\\ p_{31}\end{array}\right]\\ \left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\\ T_{mv}\left[\begin{array}{c}-{\mathrm{\Omega }}_t{a}_1\\ -{\mathrm{\Omega }}_t{a}_2\\ -{\mathrm{\Omega }}_t{a}_3\end{array}\right]\\ 0\end{array}\right]$$

$$\left[\begin{array}{c}\left[\begin{array}{c}0\\ 1\\ 0\end{array}\right]\\ \left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\\ \left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\\ T_{mv}\left[\begin{array}{c}0\\ 0\\ p_{32}\end{array}\right]\\ \left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\\ T_{mv}\left[\begin{array}{c}-{\mathrm{\Omega }}_t{b}_1\\ -{\mathrm{\Omega }}_t{b}_2\\ -{\mathrm{\Omega }}_t{b}_3\end{array}\right]\\ 0\end{array}\right]$$

$$\left[\begin{array}{c}\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]\\ \left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\\ \left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\\ T_{mv}\left[\begin{array}{c}0\\ 0\\ p_{33}\end{array}\right]\\ \left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\\ \left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\\ 0\end{array}\right]$$

Axe 8 : v ₈ Axe 9 : v ₉ Axe 10 : v ₁₀ $$\left[\begin{array}{c}\left[\begin{array}{c}\mathrm{\Delta }x\\ \mathrm{\Delta }y\\ \mathrm{\Delta }z\end{array}\right]\\ \left[\begin{array}{c}\mathrm{\Delta }{V}_x\\ \mathrm{\Delta }{V}_y\\ \mathrm{\Delta }{V}_z\end{array}\right]\\ \left[\begin{array}{c}{\mathrm{\phi }}_x\\ {\mathrm{\phi }}_y\\ {\mathrm{\phi }}_z\end{array}\right]\\ \left[\begin{array}{c}\mathrm{\Delta }{b}_{xm}\\ \mathrm{\Delta }{b}_{ym}\\ \mathrm{\Delta }{b}_{zm}\end{array}\right]\\ \left[\begin{array}{c}\mathrm{\Delta }{K}_{xm}\\ \mathrm{\Delta }{K}_{ym}\\ \mathrm{\Delta }{K}_{zm}\end{array}\right]\\ \left[\begin{array}{c}\mathrm{\Delta }{d}_{xm}\\ \mathrm{\Delta }{d}_{ym}\\ \mathrm{\Delta }{d}_{zm}\end{array}\right]\\ \mathrm{\Delta \alpha }\end{array}\right]$$

$$\left[\begin{array}{c}\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\\ \left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\\ \left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\\ \left[\begin{array}{c}-g{T}_{mv}^{13}\\ 0\\ 0\end{array}\right]\\ \left[\begin{array}{c}1\\ 0\\ 0\end{array}\right]\\ \left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\\ 0\end{array}\right]$$

$$\left[\begin{array}{c}\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\\ \left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\\ \left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\\ \left[\begin{array}{c}0\\ -g{T}_{mv}^{23}\\ 0\end{array}\right]\\ \left[\begin{array}{c}0\\ 1\\ 0\end{array}\right]\\ \left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\\ 0\end{array}\right]$$

$$\left[\begin{array}{c}\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\\ \left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\\ \left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\\ \left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\\ \left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\\ T_{mv}\left[\begin{array}{c}{\mathrm{\Omega }}_t{T}_{vt}^{21}\\ \mathrm{-}{\mathrm{\Omega }}_t{T}_{vt}^{11}\\ 0\end{array}\right]\\ 1\end{array}\right]$$

The unobservable axes are as follows: Axis 1: v ₁ Axis 2: v ₂ Axis 3: v ₃ Axis 4: v ₄ $$\left[\begin{array}{c}\left[\begin{array}{c}\mathrm{\Delta }x\\ \mathrm{\Delta }\mathrm{there}\\ \mathrm{\Delta }z\end{array}\right]\\ \left[\begin{array}{c}\mathrm{\Delta }{V}_x\\ \mathrm{\Delta }{V}_{\mathrm{there}}\\ \mathrm{\Delta }{V}_z\end{array}\right]\\ \left[\begin{array}{c}{\mathrm{\phi }}_x\\ {\mathrm{\phi }}_{\mathrm{there}}\\ {\mathrm{\phi }}_z\end{array}\right]\\ \left[\begin{array}{c}\mathrm{\Delta }{b}_{xm}\\ \mathrm{\Delta }{b}_{\mathrm{there}m}\\ \mathrm{\Delta }{b}_{zm}\end{array}\right]\\ \left[\begin{array}{c}\mathrm{\Delta }{K}_{xm}\\ \mathrm{\Delta }{K}_{\mathrm{there}m}\\ \mathrm{\Delta }{K}_{zm}\end{array}\right]\\ \left[\begin{array}{c}\mathrm{\Delta }{d}_{xm}\\ \mathrm{\Delta }{d}_{\mathrm{there}m}\\ \mathrm{\Delta }{d}_{zm}\end{array}\right]\\ \mathrm{\Delta \alpha }\end{array}\right]$$

$$\left[\begin{array}{c}\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\\ \left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\\ \left[\begin{array}{c}1\\ 0\\ 0\end{array}\right]\\ T_{mv}\left[\begin{array}{c}0\\ -\mathrm{boy\; Wut}\\ 0\end{array}\right]\\ \left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\\ T_{mv}\left[\begin{array}{c}0\\ -{\mathrm{vs}}_3\\ {\mathrm{vs}}_2\end{array}\right]\\ 0\end{array}\right]$$

$$\left[\begin{array}{c}\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\\ \left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\\ \left[\begin{array}{c}0\\ 1\\ 0\end{array}\right]\\ T_{mv}\left[\begin{array}{c}\mathrm{boy\; Wut}\\ 0\\ 0\end{array}\right]\\ \left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\\ T_{mv}\left[\begin{array}{c}{\mathrm{vs}}_3\\ 0\\ -{\mathrm{vs}}_1\end{array}\right]\\ 0\end{array}\right]$$

$$\left[\begin{array}{c}\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\\ \left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\\ \left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]\\ \left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\\ \left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\\ T_{mv}\left[\begin{array}{c}-{\mathrm{vs}}_2\\ {\mathrm{vs}}_1\\ 0\end{array}\right]\\ 0\end{array}\right]$$

$$\left[\begin{array}{c}\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\\ \left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\\ \left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\\ \left[\begin{array}{c}0\\ 0\\ -\mathrm{boy\; Wut}{T}_{mv}^{33}\end{array}\right]\\ \left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]\\ \left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\\ 0\end{array}\right]$$

Axis 5: v ₅ Axis 6: v ₆ Axis 7: v ₇ $$\left[\begin{array}{c}\left[\begin{array}{c}\mathrm{\Delta }x\\ \mathrm{\Delta }\mathrm{there}\\ \mathrm{\Delta }z\end{array}\right]\\ \left[\begin{array}{c}\mathrm{\Delta }{V}_x\\ \mathrm{\Delta }{V}_{\mathrm{there}}\\ \mathrm{\Delta }{V}_z\end{array}\right]\\ \left[\begin{array}{c}{\mathrm{\phi }}_x\\ {\mathrm{\phi }}_{\mathrm{there}}\\ {\mathrm{\phi }}_z\end{array}\right]\\ \left[\begin{array}{c}\mathrm{\Delta }{b}_{xm}\\ \mathrm{\Delta }{b}_{\mathrm{there}m}\\ \mathrm{\Delta }{b}_{zm}\end{array}\right]\\ \left[\begin{array}{c}\mathrm{\Delta }{K}_{xm}\\ \mathrm{\Delta }{K}_{\mathrm{there}m}\\ \mathrm{\Delta }{K}_{zm}\end{array}\right]\\ \left[\begin{array}{c}\mathrm{\Delta }{d}_{xm}\\ \mathrm{\Delta }{d}_{\mathrm{there}m}\\ \mathrm{\Delta }{d}_{zm}\end{array}\right]\\ \mathrm{\Delta \alpha }\end{array}\right]$$

$$\left[\begin{array}{c}\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right]\\ \left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\\ \left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\\ T_{mv}\left[\begin{array}{c}0\\ 0\\ p_{31}\end{array}\right]\\ \left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\\ T_{mv}\left[\begin{array}{c}-{\mathrm{\Omega }}_t{\mathrm{at}}_1\\ -{\mathrm{\Omega }}_t{\mathrm{at}}_2\\ -{\mathrm{\Omega }}_t{\mathrm{at}}_3\end{array}\right]\\ 0\end{array}\right]$$

$$\left[\begin{array}{c}\left[\begin{array}{c}0\\ 1\\ 0\end{array}\right]\\ \left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\\ \left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\\ T_{mv}\left[\begin{array}{c}0\\ 0\\ p_{32}\end{array}\right]\\ \left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\\ T_{mv}\left[\begin{array}{c}-{\mathrm{\Omega }}_t{b}_1\\ -{\mathrm{\Omega }}_t{b}_2\\ -{\mathrm{\Omega }}_t{b}_3\end{array}\right]\\ 0\end{array}\right]$$

$$\left[\begin{array}{c}\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]\\ \left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\\ \left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\\ T_{mv}\left[\begin{array}{c}0\\ 0\\ p_{33}\end{array}\right]\\ \left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\\ \left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\\ 0\end{array}\right]$$

Axis 8: v ₈ Axis 9: v ₉ Axis 10: v ₁₀ $$\left[\begin{array}{c}\left[\begin{array}{c}\mathrm{\Delta }x\\ \mathrm{\Delta }\mathrm{there}\\ \mathrm{\Delta }z\end{array}\right]\\ \left[\begin{array}{c}\mathrm{\Delta }{V}_x\\ \mathrm{\Delta }{V}_{\mathrm{there}}\\ \mathrm{\Delta }{V}_z\end{array}\right]\\ \left[\begin{array}{c}{\mathrm{\phi }}_x\\ {\mathrm{\phi }}_{\mathrm{there}}\\ {\mathrm{\phi }}_z\end{array}\right]\\ \left[\begin{array}{c}\mathrm{\Delta }{b}_{xm}\\ \mathrm{\Delta }{b}_{\mathrm{there}m}\\ \mathrm{\Delta }{b}_{zm}\end{array}\right]\\ \left[\begin{array}{c}\mathrm{\Delta }{K}_{xm}\\ \mathrm{\Delta }{K}_{\mathrm{there}m}\\ \mathrm{\Delta }{K}_{zm}\end{array}\right]\\ \left[\begin{array}{c}\mathrm{\Delta }{d}_{xm}\\ \mathrm{\Delta }{d}_{\mathrm{there}m}\\ \mathrm{\Delta }{d}_{zm}\end{array}\right]\\ \mathrm{\Delta \alpha }\end{array}\right]$$

$$\left[\begin{array}{c}\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\\ \left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\\ \left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\\ \left[\begin{array}{c}-\mathrm{boy\; Wut}{T}_{mv}^{13}\\ 0\\ 0\end{array}\right]\\ \left[\begin{array}{c}1\\ 0\\ 0\end{array}\right]\\ \left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\\ 0\end{array}\right]$$

$$\left[\begin{array}{c}\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\\ \left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\\ \left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\\ \left[\begin{array}{c}0\\ -\mathrm{boy\; Wut}{T}_{mv}^{23}\\ 0\end{array}\right]\\ \left[\begin{array}{c}0\\ 1\\ 0\end{array}\right]\\ \left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\\ 0\end{array}\right]$$

$$\left[\begin{array}{c}\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\\ \left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\\ \left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\\ \left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\\ \left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\\ T_{mv}\left[\begin{array}{c}{\mathrm{\Omega }}_t{T}_{vt}^{21}\\ \mathrm{-}{\mathrm{\Omega }}_t{T}_{vt}^{11}\\ 0\end{array}\right]\\ 1\end{array}\right]$$

APPENDIX 3: filter constraint

Matrix deviations are applied to the transition and observation matrices at each Kalman cycle based on a model of q unobservable axes v _{1 , k} ( X _{H, k} ), v _{2, k} ( X _{H, k} ), ... v _{q, k} ( X _{H, k} ) obtained at each cycle k according to functions of the global state X _{H, k} corresponding to the linearization point of the observation matrix at the Kalman period.

proche de φ _k→k+1(X̂_φ,k ), et tel que pour i compris entre 1 et q : $${\widehat{v}}_{i,k+1}\left({\widehat{X}}_{H,k+1}\right)={\phi }_{k\to k+1}^{*}\left({\widehat{X}}_{\phi ,k},{\widehat{X}}_{H,k},{\widehat{X}}_{H,k+1}\right){\widehat{v}}_{i,k}\left({\widehat{X}}_{H,k}\right),$$

minimum vis-à-vis de toutes les solutions possibles. φ _{k → k +1} ( X _{φ, k} ) is replaced by: ${\mathrm{\phi }}_{k\to k+1}^{*}\left({\widehat{X}}_{\phi ,k},{\widehat{X}}_{H,k},{\widehat{X}}_{H,k+1}\right)={\mathrm{\phi }}_{k\to k+1}\left({\widehat{X}}_{\phi ,k}\right)+\mathrm{\delta }{\mathrm{\phi }}_{k\to k+1}\left({\widehat{X}}_{\phi ,k},{\widehat{X}}_{H,k},{\widehat{X}}_{H,k+1}\right)$

close to φ _{k → k +1} ( X _{φ, k} ), and such that for i between 1 and q: $${\widehat{v}}_{i,k+1}\left({\widehat{X}}_{H,k+1}\right)={\phi }_{k\to k+1}^{*}\left({\widehat{X}}_{\phi ,k},{\widehat{X}}_{H,k},{\widehat{X}}_{H,k+1}\right){\widehat{v}}_{i,k}\left({\widehat{X}}_{H,k}\right),$$

minimum of all possible solutions.

proche de H_k (X̂_H,k ), et tel que pour i compris entre 1 et q : $$H_k^{*}\left({\widehat{X}}_{H,k}\right)=H_k\left({\widehat{X}}_{H,k}\right)+\delta H_K\left({\widehat{X}}_{H,k}\right)$$

minimum vis-à-vis de toutes les solutions possibles. H _k ( X _{H, k} ) is replaced by $H_k^{*}\left({\widehat{X}}_{H,k}\right)=H_k\left({\widehat{X}}_{H,k}\right)+\delta H_K\left({\widehat{X}}_{H,k}\right)$

close to H _k ( X _{H, k} ), and such that for i between 1 and q: $$H_k^{*}\left({\widehat{X}}_{H,k}\right)=H_k\left({\widehat{X}}_{H,k}\right)+\delta H_K\left({\widehat{X}}_{H,k}\right)$$

minimum of all possible solutions.

The matrix standard considered is the Froboenius standard.

et $H_k^{*}\left({\widehat{X}}_{H,k}\right)$

sont alors utilisées dans l'étape de propagation de la matrice de covariance, et du calcul du gain de Kalman.Matrices ${\mathrm{\phi }}_{k\to k+1}^{*}\left({\widehat{X}}_{\phi ,k},{\widehat{X}}_{H,k},{\widehat{X}}_{H,k+1}\right)$

and $H_k^{*}\left({\widehat{X}}_{H,k}\right)$

are then used in the propagation step of the covariance matrix, and the calculation of the Kalman gain.

est noté ${\phi }_{k\to k+1}^{*}$

δφ _k→k+1(X̂_H,k ,X̂ _H,k+1) est noté δφ _k→k+1 $H_k^{*}\left({\widehat{X}}_{H,k}\right)$

est noté $H_k^{*}$

δH_k (X̂_H,k ) est noté δH_k We now adopt the following notations: ${\phi }_{k\to k+1}^{*}\left({\widehat{X}}_{\phi ,k},{\widehat{X}}_{H,k},{\widehat{X}}_{H,k+1}\right)$

is noted ${\phi }_{k\to k+1}^{*}$

δφ _{k → k + 1} (X _{H, K,} X _{H, k + 1)} is denoted δφ _{k → k +1} $H_k^{*}\left({\widehat{X}}_{H,k}\right)$

is noted $H_k^{*}$

δH _k ( X _{H, k} ) is denoted δH _k

$$\delta H_k={\displaystyle \sum _{i=1}^q\delta H_{i,k}}$$

Et, pour i compris entre 1 et q : $$\delta {\phi }_{i,k-1\to k}=\frac{1}{{\widehat{v}}_{i,k-1}^T{\widehat{v}}_{i,k-1}}\left({\widehat{v}}_{i,k/k-1}-{\phi }_{k-1\to k}{\widehat{v}}_{i,k-1}\right){\widehat{v}}_{i,k-1}^T$$

$$\delta H_{i,k}=-\frac{1}{{\widehat{v}}_{i,k}^T{\widehat{v}}_{i,k}}\left(H_k{\widehat{v}}_{i,k}\right){\widehat{v}}_{i,k}^T$$

La démonstration de ces relations est donnée en annexe 8 selon une méthode géométrique.
Résolution des équations 1 et 2 pour une base non orthogonale, et pour q quelconque :
Cette base définit un modèle de l'espace vectoriel non observable : Toute combinaison linéaire des vecteurs de base est non observable.Resolution of equations 1 and 2 in the case where the unobservable vectors modeled v _{1, k} , v _{2 , k} , ... v _{q, k} are orthogonal to each other at each instant k:
The total matrix difference of the transition matrix corresponds to the sum of q independent matrix deviations. It is the same for the matrix gap of the observation matrix. $$\delta {\phi }_{k-1\to }={\displaystyle \underset{i=1}{\overset{q}{\Sigma }}\delta {\phi }_{i,k-1\to k}}$$

$$\delta H_k={\displaystyle \underset{i=1}{\overset{q}{\Sigma }}\delta H_{i,k}}$$

And, for i between 1 and q: $$\delta {\phi }_{i,k-1\to k}=\frac{1}{{\widehat{v}}_{i,k-1}^T{\widehat{v}}_{i,k-1}}\left({\widehat{v}}_{i,k/k-1}-{\phi }_{k-1\to k}{\widehat{v}}_{i,k-1}\right){\widehat{v}}_{i,k-1}^T$$

$$\delta H_{i,k}=-\frac{1}{{\widehat{v}}_{i,k}^T{\widehat{v}}_{i,k}}\left(H_k{\widehat{v}}_{i,k}\right){\widehat{v}}_{i,k}^T$$

The proof of these relations is given in appendix 8 according to a geometrical method.
Solving equations 1 and 2 for a non-orthogonal basis, and for any q:
This base defines a model of the unobservable vector space: Any linear combination of the basic vectors is unobservable.

The image by the observation matrix adjusted for any linear combination of these vectors is therefore zero. It is therefore possible to orthogonalize the base, and then to apply the previous method to calculate the matrix difference of the observation matrix.

On the other hand, the image by the adjusted transition matrix of a linear combination of unobservable vectors corresponds to this same linear combination applied to the images by the same transition matrix of each of the unobservable starting vectors. It is therefore possible to determine the linear combination that makes it possible to orthogonalize the base of unobservable vectors at cycle k-1, and to use this same combination to transform the unobservable basic model to cycle k. This makes it possible to apply the previous method to calculate the matrix difference of the transition matrix.

APPENDIX 4: Weighted Metrics

Scalars m _φ and m _H called "metrics" accompany the calculated adjustment matrices and give an indication of the relative amplitude of the adjustment, the aim being to modify as little as possible the transition and observation matrices.

et $\frac{\Vert \delta H_k\Vert }{\Vert H_k\Vert }$

ou des constituants élémentaires $\frac{\Vert \delta {\phi }_{i,k-1\to k}\Vert }{\Vert {\phi }_{i,k-1\to k}\Vert }$

et $\frac{\Vert \delta H_{i,k}\Vert }{\Vert H_k\Vert },$

ces normes pouvant être calculées de la même façon que celle d'un vecteur en rassemblant tous les coefficients de la matrice en une seule colonne.
Des exemples sont donnés ci-dessous : $$m_{\phi }=\frac{\parallel \delta {\phi }_{k-1\to k}\parallel }{\parallel {\phi }_{k-1\to k}\parallel }\mathrm{ou}{m}_{\phi }={\left(\frac{\parallel \delta {\phi }_{k-1\to k}\parallel }{\parallel {\phi }_{k-1\to k}\parallel }\right)}^2\mathrm{ou}{m}_{\phi }={\displaystyle \sum _{i=1}^N\frac{\parallel \delta {\phi }_{i,k-1\to k}\parallel }{\parallel {\phi }_{k-1\to k}\parallel }}$$

$$m_H=\frac{\parallel \delta H_k\parallel }{\parallel H_k\parallel }\mathrm{ou}{m}_H={\left(\frac{\parallel \delta H_k\parallel }{\parallel H_k\parallel }\right)}^2\mathrm{ou}{m}_H={\displaystyle \sum _{i=1}^N\frac{\parallel \delta H_{i,k}\parallel }{\parallel H_k\parallel }}$$

Different methods are possible to compare ∥ δφ _{k -1 → k} ∥ with ∥ φ _{k -1 → k} ∥, and ∥ δH _k ∥ with ∥ H _k ∥. The goal is to have a growing function of $\frac{\Vert \delta {\phi }_{k-1\to k}\Vert }{\Vert {\phi }_{k-1\to k}\Vert }$

and $\frac{\Vert \delta H_k\Vert }{\Vert H_k\Vert }$

or elementary constituents $\frac{\Vert \delta {\phi }_{i,k-1\to k}\Vert }{\Vert {\phi }_{i,k-1\to k}\Vert }$

and $\frac{\Vert \delta H_{i,k}\Vert }{\Vert H_k\Vert },$

these norms can be calculated in the same way as that of a vector by gathering all the coefficients of the matrix in a single column.
Examples are given below: $$m_{\phi }=\frac{\parallel \delta {\phi }_{k-1\to k}\parallel }{\parallel {\phi }_{k-1\to k}\parallel }\mathrm{or}{m}_{\phi }={\left(\frac{\parallel \delta {\phi }_{k-1\to k}\parallel }{\parallel {\phi }_{k-1\to k}\parallel }\right)}^2\mathrm{or}{m}_{\phi }={\displaystyle \underset{i=1}{\overset{\mathrm{NOT}}{\Sigma }}\frac{\parallel \delta {\phi }_{i,k-1\to k}\parallel }{\parallel {\phi }_{k-1\to k}\parallel }}$$

$$m_H=\frac{\parallel \delta H_k\parallel }{\parallel H_k\parallel }\mathrm{or}{m}_H={\left(\frac{\parallel \delta H_k\parallel }{\parallel H_k\parallel }\right)}^2\mathrm{or}{m}_H={\displaystyle \underset{i=1}{\overset{\mathrm{NOT}}{\Sigma }}\frac{\parallel \delta H_{i,k}\parallel }{\parallel H_k\parallel }}$$

et $m_H=\frac{\Vert g_H\left(\delta H_k\right)\Vert }{\Vert g_H\left(H_k\right)\Vert }$

Variants are possible by using weighting functions g _φ and g _H to balance the weight of the different components in the calculation of matrix standards. The same examples as the previous ones can be taken again taking into account these functions.
For example : $m_{\phi }=\frac{\Vert {\mathrm{boy\; Wut}}_{\phi }\left(\delta {\phi }_{k-1\to k}\right)\Vert }{\Vert {\mathrm{boy\; Wut}}_{\phi }\left({\phi }_{k-1\to k}\right)\Vert }$

and $m_H=\frac{\Vert {\mathrm{boy\; Wut}}_H\left(\delta H_k\right)\Vert }{\Vert {\mathrm{boy\; Wut}}_H\left(H_k\right)\Vert }$

A possible realization of these weighting functions consists in first constructing a weighting function f of a state vector and then deducing g _φ and g _H.

Thus by this method, the vector U = [ u ₁ u ₂ ... u _L ] ^T is associated by a weighting function f with a vector U '= f ( U ) = [ a ₁ a ₂ ... a _L ] U = [ a ₁ u ₁ a ₂ u ₂ ... a _L u _L ] ^T where the real coefficients a ₁ , a ₂ , ... a _L form the weighting coefficients. These then make it possible to weight the transition matrix as follows: $${\mathrm{boy\; Wut}}_{\phi }\left(\left[\begin{array}{cccc}{t}_{11}& t_{12}& \cdots & {\mathrm{<figure-callout\; id="1"\; label="t"\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">t</figure-callout>}}_1{}_{\mathrm{The}}\\ t_{21}& t_{22}& & {\mathrm{<figure-callout\; id="2"\; label="t"\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">t</figure-callout>}}_2{}_{\mathrm{The}}\\ \cdots & & \cdots & \\ t_{\mathrm{The}1}& t_{\mathrm{The}2}& & t_{\mathrm{The}\mathrm{The}}\end{array}\right]\right)=\left[\begin{array}{cccc}\frac{{\mathrm{at}}_1}{{\mathrm{at}}_1}{t}_{11}& \frac{{\mathrm{at}}_2}{{\mathrm{at}}_2}{t}_{12}& \cdots & \frac{{\mathrm{at}}_1}{{\mathrm{at}}_{\mathrm{<figure-callout\; id="1"\; label="NOT\; t"\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">NOT\; </figure-callout>}}}{t}_{}{}_1{}_{\mathrm{The}}\\ \frac{{\mathrm{at}}_2}{{\mathrm{at}}_1}{t}_{21}& \frac{{\mathrm{at}}_2}{{\mathrm{at}}_2}{t}_{22}& & \frac{{\mathrm{at}}_2}{{\mathrm{at}}_{\mathrm{<figure-callout\; id="2"\; label="NOT\; t"\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">NOT\; </figure-callout>}}}{t}_{}{}_2{}_{\mathrm{The}}\\ \cdots & & \cdots & \\ \frac{{\mathrm{at}}_{\mathrm{The}}}{{\mathrm{at}}_1}{t}_{\mathrm{The}1}& \frac{{\mathrm{at}}_{\mathrm{The}}}{{\mathrm{at}}_2}{t}_{\mathrm{The}2}& & \frac{{\mathrm{at}}_{\mathrm{The}}}{{\mathrm{at}}_{\mathrm{The}}}{t}_{\mathrm{The}\mathrm{The}}\end{array}\right]$$

U = [u ₁ u ₂ ... u_L ] ^T et V = [v ₁ v ₂ ... v_L ] ^T On a $g_{\phi }\left(\phi \right)f\left(U\right)=\left[\begin{array}{cccc}\frac{a_1}{a_1}{t}_{11}& \frac{a_1}{a_2}{t}_{12}& \dots & \frac{a_1}{a_L}{t}_{1L}\\ \frac{a_2}{a_1}{t}_{21}& \frac{a_2}{a_2}{t}_{22}& & \frac{a_2}{a_L}{t}_{2L}\\ \dots & & \dots & \\ \frac{a_L}{a_1}{t}_{L1}& \frac{a_L}{a_2}{t}_{L2}& & \frac{a_L}{a_L}{t}_{LL}\end{array}\right]\left[\begin{array}{c}{a}_1{u}_1\\ a_2{u}_2\\ \cdots \\ a_L{u}_L\end{array}\right]=\left[\begin{array}{cccc}{a}_1{t}_{11}& a_1{t}_{12}& \cdots & a_1{t}_{1L}\\ a_2{t}_{21}& a_2{t}_{22}& & a_2{t}_{2L}\\ \cdots & & \cdots & \\ a_L{t}_{L1}& a_L{t}_{L2}& & a_L{t}_{LL}\end{array}\right]\left[\begin{array}{c}{u}_1\\ u_2\\ \cdots \\ u_L\end{array}\right]$

C'est-à-dire $g_{\phi }\left(\phi \right)f\left(U\right)=\left[\begin{array}{cccc}{a}_1& a_2& \dots & a_L\end{array}\right]\left[\begin{array}{cccc}{t}_{11}& t_{12}& \dots & t_{1L}\\ t_{21}& t_{22}& & t_{2L}\\ \dots & & \dots & \\ t_{L1}& t_{L2}& & t_{LL}\end{array}\right]\left[\begin{array}{c}{u}_1\\ u_2\\ \dots \\ u_L\end{array}\right]=\left[\begin{array}{cccc}{a}_1& a_2& \dots & a_L\end{array}\right]\phi U$

Donc g_φ (φ)f(U)=[a ₁ a ₂ ... a_L ]V
Donc f(V) = g_φ (φ)f(U)
De façon similaire $g_H\left(\left[\begin{array}{cccc}{t}_{11}& t_{12}& \cdots & t_{1L}\\ t_{21}& t_{22}& & t_{2L}\\ \cdots & & \cdots & \\ & & & \end{array}\right]\right)=\left[\begin{array}{cccc}\frac{1}{a_1}{t}_{11}& \frac{1}{a_2}{t}_{12}& \cdots & \frac{1}{a_L}{t}_{1L}\\ \frac{1}{a_1}{t}_{21}& \frac{1}{a_{22}}{t}_{22}& & \frac{1}{a_L}{t}_{2L}\\ \cdots & & \cdots & \\ & & & \end{array}\right]$

Because if V = φU then f ( V ) = g _φ ( φ ) f ( U ) and vice versa. These are 2 equivalent ways of writing the relationship between U and V, the second way of calculating norms in a balanced way.
Indeed, by posing $\phi =\left[\begin{array}{cccc}{t}_{11}& t_{12}& \cdots & t_{1\mathrm{The}}\\ t_{21}& t_{22}& & t_{2\mathrm{The}}\\ \cdots & & \cdots & \\ t_{\mathrm{The}1}& t_{\mathrm{The}2}& & t_{\mathrm{The}\mathrm{The}}\end{array}\right],$

U = [ u ₁ u ₂ ... u _L ] ^T and V = [ v ₁ v ₂ ... v _L ] ^T We have ${\mathrm{boy\; Wut}}_{\phi }\left(\phi \right)f\left(U\right)=\left[\begin{array}{cccc}\frac{{\mathrm{at}}_1}{{\mathrm{at}}_1}{t}_{11}& \frac{{\mathrm{at}}_1}{{\mathrm{at}}_2}{t}_{12}& ...& \frac{{\mathrm{at}}_1}{{\mathrm{at}}_{\mathrm{The}}}{t}_{1\mathrm{The}}\\ \frac{{\mathrm{at}}_2}{{\mathrm{at}}_1}{t}_{21}& \frac{{\mathrm{at}}_2}{{\mathrm{at}}_2}{t}_{22}& & \frac{{\mathrm{at}}_2}{{\mathrm{at}}_{\mathrm{The}}}{t}_{2\mathrm{The}}\\ ...& & ...& \\ \frac{{\mathrm{at}}_{\mathrm{The}}}{{\mathrm{at}}_1}{t}_{\mathrm{The}1}& \frac{{\mathrm{at}}_{\mathrm{The}}}{{\mathrm{at}}_2}{t}_{\mathrm{The}2}& & \frac{{\mathrm{at}}_{\mathrm{The}}}{{\mathrm{at}}_{\mathrm{The}}}{t}_{\mathrm{The}\mathrm{The}}\end{array}\right]\left[\begin{array}{c}{\mathrm{at}}_1{u}_1\\ {\mathrm{at}}_2{u}_2\\ \cdots \\ {\mathrm{at}}_{\mathrm{The}}{u}_{\mathrm{The}}\end{array}\right]=\left[\begin{array}{cccc}{\mathrm{at}}_1{t}_{11}& {\mathrm{at}}_1{t}_{12}& \cdots & {\mathrm{at}}_1{t}_{1\mathrm{The}}\\ {\mathrm{at}}_2{t}_{21}& {\mathrm{at}}_2{t}_{22}& & {\mathrm{at}}_2{t}_{2\mathrm{The}}\\ \cdots & & \cdots & \\ {\mathrm{at}}_{\mathrm{The}}{t}_{\mathrm{The}1}& {\mathrm{at}}_{\mathrm{The}}{t}_{\mathrm{The}2}& & {\mathrm{at}}_{\mathrm{The}}{t}_{\mathrm{The}\mathrm{The}}\end{array}\right]\left[\begin{array}{c}{u}_1\\ u_2\\ \cdots \\ u_{\mathrm{The}}\end{array}\right]$

That is to say ${\mathrm{boy\; Wut}}_{\phi }\left(\phi \right)f\left(U\right)=\left[\begin{array}{cccc}{\mathrm{at}}_1& {\mathrm{at}}_2& ...& {\mathrm{at}}_{\mathrm{The}}\end{array}\right]\left[\begin{array}{cccc}{t}_{11}& t_{12}& ...& t_{1\mathrm{The}}\\ t_{21}& t_{22}& & t_{2\mathrm{The}}\\ ...& & ...& \\ t_{\mathrm{The}1}& t_{\mathrm{The}2}& & t_{\mathrm{The}\mathrm{The}}\end{array}\right]\left[\begin{array}{c}{u}_1\\ u_2\\ ...\\ u_{\mathrm{The}}\end{array}\right]=\left[\begin{array}{cccc}{\mathrm{at}}_1& {\mathrm{at}}_2& ...& {\mathrm{at}}_{\mathrm{The}}\end{array}\right]\phi U$

So g _φ ( φ ) f ( U ) = [ a ₁ to ₂ ... a _L ] V
So f ( V ) = g _φ ( φ ) f ( U )
In the same way ${\mathrm{boy\; Wut}}_H\left(\left[\begin{array}{cccc}{t}_{11}& t_{12}& \cdots & {\mathrm{<figure-callout\; id="1"\; label="t"\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">t</figure-callout>}}_{1\mathrm{The}}\\ t_{21}& t_{22}& & {\mathrm{<figure-callout\; id="2"\; label="t"\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">t</figure-callout>}}_{2\mathrm{The}}\\ \cdots & & \cdots & \\ & & & \end{array}\right]\right)=\left[\begin{array}{cccc}\frac{1}{{\mathrm{at}}_1}{t}_{11}& \frac{1}{{\mathrm{at}}_2}{t}_{12}& \cdots & \frac{1}{{\mathrm{at}}_{\mathrm{The}}}{\mathrm{<figure-callout\; id="1"\; label="t"\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">t</figure-callout>}}_{1\mathrm{The}}\\ \frac{1}{{\mathrm{at}}_1}{t}_{21}& \frac{1}{{\mathrm{at}}_{22}}{t}_{22}& & \frac{1}{{\mathrm{at}}_{\mathrm{The}}}{\mathrm{<figure-callout\; id="2"\; label="t"\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">t</figure-callout>}}_{2\mathrm{The}}\\ \cdots & & \cdots & \\ & & & \end{array}\right]$

APPENDIX 5: Concept of dynamic system and application to navigation

It is a system of deterministic equations with an evolution equation and an observation equation. The evolution equation describes a state vector according to its previous state and a command. The observation equation provides a scalar or vector dependent on the state vector. These equations can be non-linear. There are continuous time systems and discrete time systems.

Continuous time system: $\{\begin{array}{l}\dot{X}\left(t\right)=f\left(X\left(t\right),u\left(t\right)\right)\\ z\left(t\right)=h\left(X\left(t\right)\right)\end{array}$

Où X(t) et x_k désignent l'état du système, u(t) et u_k désignent la commande d'entrée, z(t) et z_k désignent l'observation à un instant continu t ou instant discret k, selon le mode de représentation continu ou discret.
Les équations de la mécanique classique de Newton permettent de décrire l'état d'un repère mobile en fonction de son accélération et de sa vitesse angulaire à tout instant. Ces équations constituent un exemple de fonction d'évolution continue, le vecteur d'état rassemblant les informations de position, de vitesse, et d'orientation du repère mobile, et le vecteur de commande étant constitué de l'accélération et la vitesse angulaire.
Dans l'état de l'art d'une centrale inertielle de navigation, les calculs ne peuvent être réalisés que de façon discrète. Il en résulte que cette fonction d'évolution continue y est modélisée sous forme d'une fonction d'évolution discrète. Différentes méthodes connues permettent de convertir les équations d'évolution continues en équations d'évolution discrètes.
La fonction d'observation modélisée dans le système dynamique à temps discret de la centrale inertielle décrit les mesures effectuées par un capteur non inertiel particulier. Plusieurs fonctions d'observation peuvent alors être mises en oeuvre successivement dans le temps en fonction des capteurs utilisés.
Le système dynamique à temps discret utilisé dans la centrale inertielle de navigation est donc un modèle le plus fidèle possible d'un autre système dynamique, à temps continu, correspondant au monde vrai. Les propriétés d'observabilité de ces deux systèmes sont donc différentes.Discrete time system: $\{\begin{array}{l}{x}_{k+1}=f\left(x_k,u_k\right)\\ z_k=h\left(x_k\right)\end{array}$

Where X ( t ) and x _k denote the state of the system, u ( t ) and u _k denote the input control, z ( t ) and z _k denote the observation at a continuous instant t or discrete instant k, according to the continuous or discrete representation mode.
The equations of Newton's classical mechanics make it possible to describe the state of a mobile marker as a function of its acceleration and its angular velocity at any moment. These equations constitute an example of a continuous evolution function, the state vector gathering the information of position, speed, and orientation of the moving marker, and the control vector consisting of the acceleration and the angular velocity.
In the state of the art of an inertial navigation unit, the calculations can be made only discretely. As a result, this continuous evolution function is modeled as a discrete evolution function. Various known methods make it possible to convert the continuous evolution equations into discrete evolution equations.
The observation function modeled in the discrete-time dynamic system of the inertial unit describes the measurements performed by a non-inertial sensor particular. Several observation functions can then be implemented successively in time depending on the sensors used.
The discrete time dynamic system used in the inertial navigation unit is therefore the most faithful model possible of another dynamic system, in continuous time, corresponding to the real world. The observability properties of these two systems are therefore different.

The observational properties of a dynamic system characterize the information on the state of the system at a given instant that can be obtained by taking into account observations made later and evolution and observation functions. These properties depend in particular on the control, that is to say the movements of the carrier in the case of the inertial navigation unit.

APPENDIX 6: Observability of a continuous time system

Consider a continuous nonlinear dynamic system that at time 0 is in a state X (0). Subsequently, the propagation function and the input control move the system into X ( t ) states , thereby defining a particular path in the state space.

où F et H sont des matrices variables dans le temps correspondant à la matrice d'évolution et la matrice d'observation du système dynamique linéarisé pour une trajectoire donnée.Since this trajectory is known, suppose that the initial state now deviates from X (0) by a small value δX (0) called the linearized state and that the input commands are the same as in the previous situation. The following states are then X ( t ) + δX ( t ) . The observation function makes it possible to collect at each step information on the current state. The problem is to know if this information from the beginning to the end of the scenario considered allows to know δX (0). The space of linearized states at time 0 is then composed of an observable subspace and an unobservable subspace. The unobservable space at time 0 characterized by the state δX (0) can then be deduced from a new dynamic system for describing the linearized states knowing the trajectory X ( t ): $$\{\begin{array}{l}\delta \dot{X}\left(t\right)=\frac{\partial f}{\partial X}\left(X\left(t\right),u\left(t\right)\right)\delta X\left(t\right)=F\left(X\left(t\right),u\left(t\right)\right)\delta X\left(t\right)\\ \delta z\left(t\right)={\left(\frac{dh}{dX}\right)}_{X\left(t\right)}\delta X\left(t\right)=H\left(X\left(t\right)\right)\delta X\left(t\right)\end{array}$$

where F and H are time-varying matrices corresponding to the evolution matrix and the observation matrix of the linearized dynamic system for a given trajectory.

pour k=1,2,3,... et t ≥ 0The unobservable space then consists of linearized states δX ( t ) such that $\frac{d^{k}z\left(t\right)}{d{t}^k}=0$

for k = 1,2,3, ... and t ≥ 0

Solutions form a vector space of linearized states. There is then a base B (0) of this space forming unobservable vectors B (0) = ( δX ₁ (0), δX ₂ (0), ... , δX _q (0)) at time 0 .

APPENDIX 7: Observability of a discrete time system

Consider a nonlinear discrete dynamic system that at time 0 is in a state X ₀ . At different successive discrete instants, the propagation function as well as the input control make the system go into states X ₁ , X ₂ , ... Knowing these successive states, suppose that the initial state now deviates from X ₀ of a small value δX ₀ called linearized state and that the input commands are the same as in the previous situation. The following successive states are then X ₁ + δX ₁ , X ₂ + δX ₂ , ... The observation function makes it possible to collect at each step information on the current state. The problem is to know if this information from the beginning to the end of the scenario considered allows to know δX ₀ . The space of linearized states at time 0 is then composed of an observable subspace and an unobservable subspace. An observer working on the principle of linearization, like the EKF, will never be able to estimate an error in the unobservable space of the linearized states.

The unobservable space at time 0 characterized by the state X ₀ can then be deduced by considering the dynamic system of linearized states δX _k , implementing the evolution function and the linearized observation function at different states. X ₀ , X ₁ , X ₂ , ... The linearized evolution function at time k is called the "transition matrix" denoted by φ _{k → k +1} ( X _k ). The linearized observation function at time k is denoted H _k ( X _k ) .

A first observation of the deviation δX ₀ around X ₀ is made using the observation matrix H ₀ ( X ₀ ) corresponding to the linearized observation function at X ₀ . A prediction of the difference δX ₁ around X ₁ is made from the difference δX ₀ around X ₀ using the transition matrix φ _{0 → 1} ( X ₀ ) corresponding to the evolution function linearized in X ₀ . A second observation of the deviation δX ₁ around X ₁ is made using the observation matrix H ₁ ( X ₁ ), etc.

L'espace non observable et l'espace observable dépendent donc de l'état de référence initial X ₀ , de la commande d'entrée (données inertielles dans le cas des centrales inertielles), et des fonctions d'évolution et d'observation.We then know that the unobservable space of the system describing the perturbation is the core of the matrix: $$M\left(X_0\right)=\left[\begin{array}{c}{H}_0\left(X_0\right)\\ H_1\left({\mathrm{<figure-callout\; id="1"\; label="X"\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">X</figure-callout>}}_1\right){\mathrm{\phi }}_{0\to 1}\left(X_0\right)\\ H_2\left({\mathrm{<figure-callout\; id="2"\; label="X"\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">X</figure-callout>}}_2\right){\mathrm{<figure-callout\; id="1"\; label="\phi "\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">\phi </figure-callout>}}_{1\to 2}\left({\mathrm{<figure-callout\; id="1"\; label="X"\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">X</figure-callout>}}_1\right){\mathrm{\phi }}_{0\to 1}\left(X_0\right)\\ \cdots \end{array}\right]$$

The unobservable space and the observable space therefore depend on the initial reference state X ₀ , the input control (inertial data in the case of inertial units), and the evolution and observation functions.

On a alors : $\{\begin{array}{l}{H}_0\left(X_0\right)v_0\left(X_0\right)=0\\ H_1\left(X_1\right)v_1\left(X_1\right)=0\\ H_2\left(X_2\right)v_2\left(X_2\right)=0\\ \dots \end{array}$

Suppose that the vector v ₀ ( X ₀ ) is part of the core of M ( X ₀ ) .
Let's ask: $\{\begin{array}{l}{v}_1\left(X_1\right)={\phi }_{0\to 1}\left(X_0\right)v_0\left(X_0\right)\\ v_2\left(X_2\right)={\mathrm{<figure-callout\; id="1"\; label="\phi "\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">\phi </figure-callout>}}_{1\to 2}\left(X_1\right)v_1\left(X_1\right)\\ v_3\left(X_3\right)={\mathrm{<figure-callout\; id="2"\; label="\phi "\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">\phi </figure-callout>}}_{2\to 3}\left(X_2\right)v_2\left({\mathrm{<figure-callout\; id="2"\; label="X"\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">X</figure-callout>}}_2\right)\\ ...\end{array},$

We then have: $\{\begin{array}{l}{H}_0\left(X_0\right)v_0\left(X_0\right)=0\\ H_1\left(X_1\right)v_1\left(X_1\right)=0\\ H_2\left(X_2\right)v_2\left(X_2\right)=0\\ ...\end{array}$

Où v_k (X_k ) forme un vecteur non observable particulier à un instant discret k.At a given moment, an observability condition is then formed for a non-observable direction by the system of equations: $$\{\begin{array}{l}{v}_{k+1}\left(X_{k+1}\right)={\phi }_{k\to k+1}\left(X_k\right)v_k\left(X_k\right)\\ H_{k+1}\left(X_{k+1}\right)v_{k+1}\left(X_{k+1}\right)=0\end{array}$$

Where v _k ( X _k ) forms a particular unobservable vector at a discrete moment k.

This condition applies at every moment and is related to the trajectory X ₀ , X ₁ , X ₂ , ...

In the EKF, this trajectory is noisy by the estimation errors. This noise tends to reduce the core of the observability matrix, and therefore to reduce the size of the unobservable space.

Avec: $\{\begin{array}{l}{H}_0\left(X_{H\mathrm{,0}}\right)v_0\left(X_{H\mathrm{,0}}\right)=0\\ H_1\left(X_{H\mathrm{,1}}\right)v_1\left(X_{H\mathrm{,1}}\right)=0\\ H_2\left(X_{H\mathrm{,2}}\right)v_2\left(X_{H\mathrm{,2}}\right)=0\\ \dots \end{array}$

Moreover, the transition and observation functions are not linearized at the same point. The sequence X ₀ , X ₁ , X ₂ , ... is replaced by a sequence X _{H, 0} , X _{φ , 1} , X _{H, 1} , X _{φ, 2} , X _{H, 2} , X _{φ, 3} ,, ...
And: $\{\begin{array}{l}{v}_1\left(X_{H\mathrm{,\; 1}}\right)={\phi }_{0\to 1}\left(X_{\phi \mathrm{,\; 0}}\right)v_0\left(X_{H\mathrm{,\; 0}}\right)\\ v_2\left({\mathrm{<figure-callout\; id="2"\; label="X\; H"\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">X\; </figure-callout>}}_H\right)={\mathrm{<figure-callout\; id="1"\; label="\phi "\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">\phi </figure-callout>}}_{1\to 2}\left(X_{\phi \mathrm{,\; 1}}\right)v_1\left(X_{H\mathrm{,\; 1}}\right)\\ v_3\left({\mathrm{<figure-callout\; id="3"\; label="X\; H"\; filenames="imgf0001.png"\; state="\{\{state\}\}">X\; </figure-callout>}}_H\right)={\mathrm{<figure-callout\; id="2"\; label="\phi "\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">\phi </figure-callout>}}_{2\to 3}\left(X_{\phi 2}\right)v_2\left({\mathrm{<figure-callout\; id="2"\; label="X\; H"\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">X\; </figure-callout>}}_H\right)\\ ...\end{array},$

With: $\{\begin{array}{l}{H}_0\left(X_{H\mathrm{,\; 0}}\right)v_0\left(X_{H\mathrm{,\; 0}}\right)=0\\ H_1\left(X_{H\mathrm{,\; 1}}\right)v_1\left(X_{H\mathrm{,\; 1}}\right)=0\\ H_2\left(X_{H2}\right)v_2\left({\mathrm{<figure-callout\; id="2"\; label="X\; H"\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">X\; </figure-callout>}}_H\right)=0\\ ...\end{array}$

The condition of observability is then written: $$\{\begin{array}{l}{v}_{k+1}\left({\widehat{X}}_{H,k+1}\right)={\phi }_{k\to k+1}\left({\widehat{X}}_{\phi ,k}\right)v_k\left({\widehat{X}}_{H,k}\right)\\ H_{k+1}\left({\widehat{X}}_{H,k+1}\right)v_{k+1}\left({\widehat{X}}_{H,k+1}\right)=0\end{array}$$

APPENDIX 8: Demonstration of the matrix fitting formula for an orthogonal vector basis

Hypothesis: p matrices U _i such that: $$\boxed{\forall \left(\mathrm{i},\mathrm{j}\right)\mathrm{tq\; i}\ne \mathrm{j}{\mathrm{,\; U}}_{\mathrm{i}}^{\mathrm{T}}{\mathrm{U}}_{\mathrm{j}}=0}$$

Let p matrices V _i of the same dimension as the U _i .

Let the matrix A.

Problem: We search for the minimum standard matrix δA such that: $$\boxed{\left(\mathrm{AT}+\mathrm{dA}\right)\left(\begin{array}{cccc}{\mathrm{U}}_1& {\mathrm{<figure-callout\; id="2"\; label="U"\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">U</figure-callout>}}_2& \cdots & {\mathrm{U}}_{\mathrm{q}}\end{array}\right)=\left(\begin{array}{cccc}{\mathrm{V}}_1& {\mathrm{<figure-callout\; id="2"\; label="V"\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">V</figure-callout>}}_2& \cdots & {\mathrm{V}}_{\mathrm{q}}\end{array}\right)}$$

The matrix A + δA can be in the form of p matrices row by column, and the vectors V _k in the form of q scalars in column: $$\left(\begin{array}{c}{\mathrm{The}}_1+\mathrm{\delta }{\mathrm{The}}_1\\ {\mathrm{The}}_2+\mathrm{\delta }{\mathrm{The}}_2\\ ...\\ {\mathrm{The}}_{\mathrm{p}}+\mathrm{\delta }{\mathrm{The}}_{\mathrm{p}}\end{array}\right)\left(\begin{array}{cccc}{\mathrm{U}}_1& {\mathrm{<figure-callout\; id="2"\; label="U"\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">U</figure-callout>}}_2& ...& {\mathrm{U}}_{\mathrm{q}}\end{array}\right)=\left(\begin{array}{cccc}{v}_{11}& v_{12}& \cdots & v_{1\mathrm{q}}\\ v_{21}& v_{22}& & v_{2\mathrm{q}}\\ ...& ...& & ...\\ {\mathrm{v}}_{\mathrm{q}1}& {\mathrm{<figure-callout\; id="2"\; label="v\; q"\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">v\; </figure-callout>}}_{\mathrm{q}}& ...& v_{\mathrm{qq}}\end{array}\right)$$

We therefore look for minimum standard δL _k such as: $$\boxed{\forall \left(\mathrm{k},\mathrm{l}\right)\in {\left\{\mathrm{1.2,}...,\mathrm{<figure-callout\; id="2"\; label="q"\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">q</figure-callout>}\right\}}^2,\left({\mathrm{The}}_{\mathrm{k}}+\mathrm{\delta }{\mathrm{The}}_{\mathrm{k}}\right){\mathrm{U}}_{\mathrm{l}}=v_{\mathrm{kl}}}$$

Geometric interpretation of the problem: The row and column matrices are associated with vectors in a vector space larger than q. We keep the same notations by adding an arrow above the variables.

Condition (1) results in: ∀ (i, j) tq i ≠ j, U _i ⊥ U _j

Condition (2) results in the following scalar product: $$\forall \left(\mathrm{k},\mathrm{l}\right)\in {\left\{\mathrm{1.2,}...,\mathrm{<figure-callout\; id="2"\; label="q"\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">q</figure-callout>}\right\}}^2,\left({\overrightarrow{\mathrm{The}}}_{\mathrm{k}}+\mathrm{\delta }{\overrightarrow{\mathrm{The}}}_{\mathrm{k}}\right){\overrightarrow{\mathrm{U}}}_{\mathrm{l}}=v_{\mathrm{kl}}$$

a pour abscisse ${\overrightarrow{\mathrm{L}}}_{\mathrm{k}}.\frac{{\overrightarrow{\mathrm{U}}}_{\mathrm{l}}}{\Vert {\overrightarrow{\mathrm{U}}}_{\mathrm{l}}\Vert }=\frac{{\overrightarrow{\mathrm{L}}}_{\mathrm{k}}\cdot {\overrightarrow{\mathrm{U}}}_{\mathrm{l}}}{\Vert {\overrightarrow{\mathrm{U}}}_{\mathrm{l}}\Vert }$

sur cet axe. C'est le cosinus directeur de L _k selon cet axe.The projection of the vector The _k on the oriented axis defined by the unit vector $\frac{{\overrightarrow{\mathrm{U}}}_{\mathrm{l}}}{\Vert {\overrightarrow{\mathrm{U}}}_{\mathrm{l}}\Vert }$

for abscissa ${\overrightarrow{\mathrm{The}}}_{\mathrm{k}}.\frac{{\overrightarrow{\mathrm{U}}}_{\mathrm{l}}}{\Vert {\overrightarrow{\mathrm{U}}}_{\mathrm{l}}\Vert }=\frac{{\overrightarrow{\mathrm{The}}}_{\mathrm{k}}\cdot {\overrightarrow{\mathrm{U}}}_{\mathrm{l}}}{\Vert {\overrightarrow{\mathrm{U}}}_{\mathrm{l}}\Vert }$

on this axis. This is the cosine director of The _k along this axis.

We search δ The _k such as the projection of The _k + δ The _k on this axis either $\frac{\left({\overrightarrow{\mathrm{The}}}_{\mathrm{k}}+\mathrm{\delta }{\overrightarrow{\mathrm{The}}}_{\mathrm{k}}\right).{\overrightarrow{\mathrm{U}}}_{\mathrm{l}}}{\Vert {\overrightarrow{\mathrm{U}}}_{\mathrm{l}}\Vert }=\frac{{\mathrm{v}}_{\mathrm{kl}}}{\Vert {\overrightarrow{\mathrm{U}}}_{\mathrm{l}}\Vert }.$

δ The _k can be decomposed according to the vectors U _i which generate a vector space of dimension q, and according to a residual component ε _k orthogonal to this space: $\boxed{\mathrm{\delta }{\overrightarrow{\mathrm{The}}}_{\mathrm{k}}={\displaystyle {\Sigma }_{\mathrm{l}=1}^{\mathrm{q}}\mathrm{\delta }{\overrightarrow{\mathrm{The}}}_{\mathrm{kl}}+{\overrightarrow{\mathrm{\epsilon }}}_{\mathrm{k}}}}$

And so $\frac{\left({\overrightarrow{\mathrm{The}}}_{\mathrm{k}}+\mathrm{\delta }{\overrightarrow{\mathrm{The}}}_{\mathrm{kl}}\right).{\overrightarrow{\mathrm{U}}}_{\mathrm{l}}}{\Vert {\overrightarrow{\mathrm{U}}}_{\mathrm{l}}\Vert }=\frac{{\mathrm{v}}_{\mathrm{kl}}}{\Vert {\overrightarrow{\mathrm{U}}}_{\mathrm{l}}\Vert }.$

The figure 7 shows that there are infinite solutions x _kl . The minimal solution corresponds to a vector δ The _kl collinear to U _l .

$$\mathrm{\delta }{\overrightarrow{\mathrm{L}}}_{\mathrm{kl}}=\left(\frac{v_{\mathrm{kl}}}{\Vert {\overrightarrow{\mathrm{U}}}_{\mathrm{l}}\Vert }-\frac{{\overrightarrow{\mathrm{L}}}_{\mathrm{k}}.{\overrightarrow{\mathrm{U}}}_{\mathrm{l}}}{\Vert {\overrightarrow{\mathrm{U}}}_{\mathrm{l}}\Vert }\right)\frac{{\overrightarrow{\mathrm{U}}}_{\mathrm{l}}}{\Vert {\overrightarrow{\mathrm{U}}}_{\mathrm{l}}\Vert }$$

$$\mathrm{\delta }{\overrightarrow{\mathrm{L}}}_{\mathrm{kl}}=\frac{1}{{\Vert {\overrightarrow{\mathrm{U}}}_{\mathrm{l}}\Vert }^2}\left(v_{\mathrm{kl}}-{\overrightarrow{\mathrm{L}}}_{\mathrm{k}}.{\overrightarrow{\mathrm{U}}}_{\mathrm{l}}\right){\overrightarrow{\mathrm{U}}}_{\mathrm{l}}$$

According to the figure, we have: $$\mathrm{\delta }{\overrightarrow{\mathrm{The}}}_{\mathrm{kl}}=\frac{v_{\mathrm{kl}}}{\Vert {\overrightarrow{\mathrm{U}}}_{\mathrm{l}}\Vert }\frac{{\overrightarrow{\mathrm{U}}}_{\mathrm{l}}}{\Vert {\overrightarrow{\mathrm{U}}}_{\mathrm{l}}\Vert }-\frac{{\overrightarrow{\mathrm{The}}}_{\mathrm{k}}.{\overrightarrow{\mathrm{U}}}_{\mathrm{l}}}{\Vert {\overrightarrow{\mathrm{U}}}_{\mathrm{l}}\Vert }\frac{{\overrightarrow{\mathrm{U}}}_{\mathrm{l}}}{\Vert {\overrightarrow{\mathrm{U}}}_{\mathrm{l}}\Vert }$$

$$\mathrm{\delta }{\overrightarrow{\mathrm{The}}}_{\mathrm{kl}}=\left(\frac{v_{\mathrm{kl}}}{\Vert {\overrightarrow{\mathrm{U}}}_{\mathrm{l}}\Vert }-\frac{{\overrightarrow{\mathrm{The}}}_{\mathrm{k}}.{\overrightarrow{\mathrm{U}}}_{\mathrm{l}}}{\Vert {\overrightarrow{\mathrm{U}}}_{\mathrm{l}}\Vert }\right)\frac{{\overrightarrow{\mathrm{U}}}_{\mathrm{l}}}{\Vert {\overrightarrow{\mathrm{U}}}_{\mathrm{l}}\Vert }$$

$$\mathrm{\delta }{\overrightarrow{\mathrm{The}}}_{\mathrm{kl}}=\frac{1}{{\Vert {\overrightarrow{\mathrm{U}}}_{\mathrm{l}}\Vert }^2}\left(v_{\mathrm{kl}}-{\overrightarrow{\mathrm{The}}}_{\mathrm{k}}.{\overrightarrow{\mathrm{U}}}_{\mathrm{l}}\right){\overrightarrow{\mathrm{U}}}_{\mathrm{l}}$$

So we have ε _k = 0 , and: $$\mathrm{\delta }{\overrightarrow{\mathrm{The}}}_{\mathrm{k}}={\displaystyle \underset{\mathrm{l}=1}{\overset{\mathrm{<figure-callout\; id="1"\; label="q"\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">q</figure-callout>}}{\Sigma }}\frac{1}{{\Vert {\overrightarrow{\mathrm{U}}}_{\mathrm{l}}\Vert }^2}\left(v_{\mathrm{kl}}-{\overrightarrow{\mathrm{The}}}_{\mathrm{k}}.{\overrightarrow{\mathrm{U}}}_{\mathrm{l}}\right){\overrightarrow{\mathrm{U}}}_{\mathrm{l}}}$$

Back in matrix notation, we have: $$\mathrm{\delta }{\mathrm{The}}_{\mathrm{k}}={\displaystyle \underset{\mathrm{l}=1}{\overset{\mathrm{q}}{\Sigma }}\frac{1}{{\mathrm{U}}_{\mathrm{l}}{}^{\mathrm{T}}{\mathrm{U}}_{\mathrm{l}}}\left(v_{\mathrm{kl}}-{\mathrm{The}}_{\mathrm{k}}.{\mathrm{U}}_{\mathrm{l}}\right){\mathrm{U}}_{\mathrm{l}}{}^{\mathrm{T}}}$$

By reforming matrices A and δA, after a few manipulations, we obtain the relation: $$\boxed{\mathrm{dA}={\displaystyle \underset{\mathrm{l}=1}{\overset{\mathrm{q}}{\Sigma }}\frac{1}{{\mathrm{U}}_{\mathrm{l}}{}^{\mathrm{T}}{\mathrm{U}}_{\mathrm{l}}}\left({\mathrm{V}}_{\mathrm{l}}-{\mathrm{AT}}_{\mathrm{l}}\right){\mathrm{U}}_{\mathrm{l}}{}^{\mathrm{T}}}}$$

Claims (8)

1. Method for estimating a navigation state with a plurality of variables of a mobile carrier according to the extended Kalman filter method, comprising the steps of:

   - acquiring measurements of at least one of the variables,

   - extended Kalman filtering (400) producing a current estimated state and a covariance matrix delimiting in the space of the navigation state a region of errors, from a preceding estimated state, an observation matrix (Φ _k-1→k ), a transition matrix (H_k ) and acquired measurements,

   the method being characterised in that it comprises a step of adjusting (310, 330) the transition matrix (H_k ) and the observation matrix (Φ _k-1→k ) prior to their use in the extended Kalman filtering, so that the adjusted matrices satisfy an observability condition that depends on at least one of the variables of the state of the carrier, the observability condition being adapted to prevent the Kalman filter from reducing the dimension of the region along at least one non-observable axis of the state space, wherein the observability condition to be satisfied by the adjusted transition and observation matrices is the nullity of the kernel of an observability matrix (M(X ₀)) which is associated therewith, and
   wherein the adjustment comprises the steps of:

   - calculating (301) at least one primary base (B ¹ _k ) of non-observable vectors from a preceding estimated state,

   - for each matrix to be adjusted (H_k, Φ _k-1→k ), calculating (306, 308) at least one matrix difference (δH_k, δΦ _k-1→k ) associated with the matrix from the primary base (B ¹ _k ) of vectors,

   - offsetting (330) each matrix to be adjusted (H_k , Φ _k-1→k ), according to the matrix difference (δH_k, δΦ _k-1→k ) that is associated therewith so as to satisfy the observability condition.
2. Method according to Claim 1, in which the extended Kalman filtering comprises the sub-steps of:

   - propagating the preceding estimated state into a predicted state by means of the adjusted transition matrix,

   - linearising in the predicted state a non-linear model so as to produce the observation matrix (Φ _k-1→k ), before adjustment,

   - adjusting the observation matrix produced by linearising.
3. Estimation method according to any one of the foregoing claims, also comprising a step of orthogonalizing (302) the primary base (B ¹ _k ) into a secondary base (B ² _k) of vectors, the matrix difference (δΦ _k-1→k ) associated with the observation matrix (Φ_k-1→k ) being calculated from the secondary base (B ² _k) of vectors.
4. Estimation method according to any one of the foregoing claims, in which the matrix difference (δΦ _i,x-1→k ) associated with the observation matrix is the sum of a plurality of elementary matrix differences (δΦ _i,k-1→k ), each elementary matrix difference (δΦ_k-1→k ) being calculated from a vector of the secondary base (B ² _k) particular thereto.
5. Estimation method according to any one of the foregoing claims, the steps whereof are repeated in successive cycles, and in which a given cycle, called current cycle, comprises the steps of:

   - storing (303) the secondary base (B ² _k) of orthogonalization vectors and coefficients, which are also produced by the orthogonalisation step, and

   - transforming (304) the primary base (B ¹ _k ) of vectors into a tertiary base (B ³ _k ) of vectors by means of orthogonalization coefficients stored during a preceding cycle,

   the matrix difference associated with the transition matrix being calculated (306) from a secondary base (B ² _k-1 ) stored during the preceding cycle and the tertiary base (B ³ _k ) calculated during the current cycle.
6. Estimation method according to the foregoing claim, wherein the matrix difference (δH_k ) associated with the transition matrix is the sum of a plurality of elementary matrix differences (δH_i,k ), each elementary matrix difference (δH_i,k ) being calculated from a vector of the secondary base (B ² _k-1 ) stored during the preceding cycle and specific to the elementary matrix difference, and from a vector of the tertiary base (B ³ _k ) calculated during the current cycle and specific to the elementary matrix difference (δH_i,x ).
7. Inertial unit (IN) comprising a plurality of sensors (CI, CC) and means (E) for estimating a navigation state of the inertial unit by executing the estimation method according to any one of the foregoing claims.
8. Computer program product comprising program code instructions for execution of the steps of the method according to any one of claims 1 to 6, when this program is executed by data-processing means (E).

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